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id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Technical_discussion" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Technical_discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Technical discussion</span> </div> </a> <button aria-controls="toc-Technical_discussion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Technical discussion subsection</span> </button> <ul id="toc-Technical_discussion-sublist" class="vector-toc-list"> <li id="toc-The_structure_of_finite_element_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_structure_of_finite_element_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The structure of finite element methods</span> </div> </a> <ul id="toc-The_structure_of_finite_element_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Illustrative_problems_P1_and_P2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Illustrative_problems_P1_and_P2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Illustrative problems P1 and P2</span> </div> </a> <ul id="toc-Illustrative_problems_P1_and_P2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Weak formulation</span> </div> </a> <ul id="toc-Weak_formulation-sublist" class="vector-toc-list"> <li id="toc-The_weak_form_of_P1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_weak_form_of_P1"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>The weak form of P1</span> </div> </a> <ul id="toc-The_weak_form_of_P1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_weak_form_of_P2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_weak_form_of_P2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>The weak form of P2</span> </div> </a> <ul id="toc-The_weak_form_of_P2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_proof_outline_of_the_existence_and_uniqueness_of_the_solution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#A_proof_outline_of_the_existence_and_uniqueness_of_the_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.3</span> <span>A proof outline of the existence and uniqueness of the solution</span> </div> </a> <ul id="toc-A_proof_outline_of_the_existence_and_uniqueness_of_the_solution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Discretization" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Discretization"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Discretization</span> </div> </a> <button aria-controls="toc-Discretization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Discretization subsection</span> </button> <ul id="toc-Discretization-sublist" class="vector-toc-list"> <li id="toc-For_problem_P1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_problem_P1"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>For problem P1</span> </div> </a> <ul id="toc-For_problem_P1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-For_problem_P2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_problem_P2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>For problem P2</span> </div> </a> <ul id="toc-For_problem_P2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Choosing_a_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Choosing_a_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Choosing a basis</span> </div> </a> <ul id="toc-Choosing_a_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Small_support_of_the_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Small_support_of_the_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Small support of the basis</span> </div> </a> <ul id="toc-Small_support_of_the_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_form_of_the_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_form_of_the_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Matrix form of the problem</span> </div> </a> <ul id="toc-Matrix_form_of_the_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_form_of_the_finite_element_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_form_of_the_finite_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>General form of the finite element method</span> </div> </a> <ul id="toc-General_form_of_the_finite_element_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Various_types_of_finite_element_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Various_types_of_finite_element_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Various types of finite element methods</span> </div> </a> <button aria-controls="toc-Various_types_of_finite_element_methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Various types of finite element methods subsection</span> </button> <ul id="toc-Various_types_of_finite_element_methods-sublist" class="vector-toc-list"> <li id="toc-AEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#AEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>AEM</span> </div> </a> <ul id="toc-AEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A-FEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A-FEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>A-FEM</span> </div> </a> <ul id="toc-A-FEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CutFEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CutFEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>CutFEM</span> </div> </a> <ul id="toc-CutFEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_finite_element_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_finite_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Generalized finite element method</span> </div> </a> <ul id="toc-Generalized_finite_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixed_finite_element_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixed_finite_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Mixed finite element method</span> </div> </a> <ul id="toc-Mixed_finite_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variable_–_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variable_–_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Variable – polynomial</span> </div> </a> <ul id="toc-Variable_–_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-hpk-FEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#hpk-FEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>hpk-FEM</span> </div> </a> <ul id="toc-hpk-FEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-XFEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#XFEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>XFEM</span> </div> </a> <ul id="toc-XFEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaled_boundary_finite_element_method_(SBFEM)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaled_boundary_finite_element_method_(SBFEM)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Scaled boundary finite element method (SBFEM)</span> </div> </a> <ul id="toc-Scaled_boundary_finite_element_method_(SBFEM)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-S-FEM" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#S-FEM"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.10</span> <span>S-FEM</span> </div> </a> <ul id="toc-S-FEM-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_element_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectral_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.11</span> <span>Spectral element method</span> </div> </a> <ul id="toc-Spectral_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Meshfree_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Meshfree_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.12</span> <span>Meshfree methods</span> </div> </a> <ul id="toc-Meshfree_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discontinuous_Galerkin_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discontinuous_Galerkin_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.13</span> <span>Discontinuous Galerkin methods</span> </div> </a> <ul id="toc-Discontinuous_Galerkin_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_element_limit_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_element_limit_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.14</span> <span>Finite element limit analysis</span> </div> </a> <ul id="toc-Finite_element_limit_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stretched_grid_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stretched_grid_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.15</span> <span>Stretched grid method</span> </div> </a> <ul id="toc-Stretched_grid_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loubignac_iteration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loubignac_iteration"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.16</span> <span>Loubignac iteration</span> </div> </a> <ul id="toc-Loubignac_iteration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Crystal_plasticity_finite_element_method_(CPFEM)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Crystal_plasticity_finite_element_method_(CPFEM)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.17</span> <span>Crystal plasticity finite element method (CPFEM)</span> </div> </a> <ul id="toc-Crystal_plasticity_finite_element_method_(CPFEM)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Virtual_element_method_(VEM)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Virtual_element_method_(VEM)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.18</span> <span>Virtual element method (VEM)</span> </div> </a> <ul id="toc-Virtual_element_method_(VEM)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Link_with_the_gradient_discretization_method" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Link_with_the_gradient_discretization_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Link with the gradient discretization method</span> </div> </a> <ul id="toc-Link_with_the_gradient_discretization_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_to_the_finite_difference_method" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Comparison_to_the_finite_difference_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Comparison to the finite difference method</span> </div> </a> <ul id="toc-Comparison_to_the_finite_difference_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_element_and_fast_fourier_transform_(FFT)_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Finite_element_and_fast_fourier_transform_(FFT)_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Finite element and fast fourier transform (FFT) methods</span> </div> </a> <ul id="toc-Finite_element_and_fast_fourier_transform_(FFT)_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Application"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Application</span> </div> </a> <ul id="toc-Application-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sonlu_elementl%C9%99r_%C3%BCsulu" title="Sonlu elementlər üsulu – Azerbaijani" lang="az" hreflang="az" data-title="Sonlu elementlər üsulu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BD%D0%B0_%D0%BA%D1%80%D0%B0%D0%B9%D0%BD%D0%B8%D1%82%D0%B5_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B8" title="Метод на крайните елементи – Bulgarian" lang="bg" hreflang="bg" data-title="Метод на крайните елементи" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_d%27elements_finits" title="Anàlisi d'elements finits – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi d'elements finits" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Metoda_kone%C4%8Dn%C3%BDch_prvk%C5%AF" title="Metoda konečných prvků – Czech" lang="cs" hreflang="cs" data-title="Metoda konečných prvků" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%B7%D8%B1%D9%8A%D9%82%D8%A9_%D9%84%D8%B9%D9%86%D8%A7%D8%B5%D8%B1_%D9%84%D9%85%D9%86%D8%AA%D9%87%D9%8A%D8%A9" title="طريقة لعناصر لمنتهية – Moroccan Arabic" lang="ary" hreflang="ary" data-title="طريقة لعناصر لمنتهية" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Finite-Elemente-Methode" title="Finite-Elemente-Methode – German" lang="de" hreflang="de" data-title="Finite-Elemente-Methode" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/L%C3%B5plike_elementide_meetod" title="Lõplike elementide meetod – Estonian" lang="et" hreflang="et" data-title="Lõplike elementide meetod" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%AD%CE%B8%CE%BF%CE%B4%CE%BF%CF%82_%CF%80%CE%B5%CF%80%CE%B5%CF%81%CE%B1%CF%83%CE%BC%CE%AD%CE%BD%CF%89%CE%BD_%CF%83%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%AF%CF%89%CE%BD" title="Μέθοδος πεπερασμένων στοιχείων – Greek" lang="el" hreflang="el" data-title="Μέθοδος πεπερασμένων στοιχείων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A9todo_de_los_elementos_finitos" title="Método de los elementos finitos – Spanish" lang="es" hreflang="es" data-title="Método de los elementos finitos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elementu_finituen_metodo" title="Elementu finituen metodo – Basque" lang="eu" hreflang="eu" data-title="Elementu finituen metodo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D9%88%D8%B4_%D8%A7%D8%AC%D8%B2%D8%A7%D8%A1_%D9%85%D8%AD%D8%AF%D9%88%D8%AF" title="روش اجزاء محدود – Persian" lang="fa" hreflang="fa" data-title="روش اجزاء محدود" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_des_%C3%A9l%C3%A9ments_finis" title="Méthode des éléments finis – French" lang="fr" hreflang="fr" data-title="Méthode des éléments finis" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%95%9C%EC%9A%94%EC%86%8C%EB%B2%95" title="유한요소법 – Korean" lang="ko" hreflang="ko" data-title="유한요소법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A4%BF%E0%A4%A4_%E0%A4%85%E0%A4%B5%E0%A4%AF%E0%A4%B5_%E0%A4%B5%E0%A4%BF%E0%A4%A7%E0%A4%BF" title="परिमित अवयव विधि – Hindi" lang="hi" hreflang="hi" data-title="परिमित अवयव विधि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Metode_elemen_hingga" title="Metode elemen hingga – Indonesian" lang="id" hreflang="id" data-title="Metode elemen hingga" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metodo_degli_elementi_finiti" title="Metodo degli elementi finiti – Italian" lang="it" hreflang="it" data-title="Metodo degli elementi finiti" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%9E%D7%A0%D7%98%D7%99%D7%9D_%D7%A1%D7%95%D7%A4%D7%99%D7%99%D7%9D" title="אלמנטים סופיים – Hebrew" lang="he" hreflang="he" data-title="אלמנטים סופיים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Baigtini%C5%B3_element%C5%B3_metodas" title="Baigtinių elementų metodas – Lithuanian" lang="lt" hreflang="lt" data-title="Baigtinių elementų metodas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A9geselemes_m%C3%B3dszer" title="Végeselemes módszer – Hungarian" lang="hu" hreflang="hu" data-title="Végeselemes módszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eindige-elementenmethode" title="Eindige-elementenmethode – Dutch" lang="nl" hreflang="nl" data-title="Eindige-elementenmethode" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E8%A6%81%E7%B4%A0%E6%B3%95" title="有限要素法 – Japanese" lang="ja" hreflang="ja" data-title="有限要素法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Finittelementmetoden" title="Finittelementmetoden – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Finittelementmetoden" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9C%E1%9E%B7%E1%9E%92%E1%9E%B8%E1%9E%A0%E1%9F%92%E1%9E%9C%E1%9F%83%E1%9E%8E%E1%9F%83%E1%9E%90%E1%9F%8D%E1%9E%A2%E1%9F%8A%E1%9F%81%E1%9E%9B%E1%9E%98%E1%9F%89%E1%9E%B7%E1%9E%93" title="វិធីហ្វៃណៃថ៍អ៊េលម៉ិន – Khmer" lang="km" hreflang="km" data-title="វិធីហ្វៃណៃថ៍អ៊េលម៉ិន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metoda_element%C3%B3w_sko%C5%84czonych" title="Metoda elementów skończonych – Polish" lang="pl" hreflang="pl" data-title="Metoda elementów skończonych" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A9todo_dos_elementos_finitos" title="Método dos elementos finitos – Portuguese" lang="pt" hreflang="pt" data-title="Método dos elementos finitos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Metoda_elementelor_finite" title="Metoda elementelor finite – Romanian" lang="ro" hreflang="ro" data-title="Metoda elementelor finite" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D1%8B%D1%85_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%BE%D0%B2" title="Метод конечных элементов – Russian" lang="ru" hreflang="ru" data-title="Метод конечных элементов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Finite_element_method" title="Finite element method – Simple English" lang="en-simple" hreflang="en-simple" data-title="Finite element method" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Met%C3%B3da_kone%C4%8Dn%C3%BDch_prvkov" title="Metóda konečných prvkov – Slovak" lang="sk" hreflang="sk" data-title="Metóda konečných prvkov" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Metoda_kon%C4%8Dnih_elementov" title="Metoda končnih elementov – Slovenian" lang="sl" hreflang="sl" data-title="Metoda končnih elementov" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Elementtimenetelm%C3%A4" title="Elementtimenetelmä – Finnish" lang="fi" hreflang="fi" data-title="Elementtimenetelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Finita_elementmetoden" title="Finita elementmetoden – Swedish" lang="sv" hreflang="sv" data-title="Finita elementmetoden" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B9%80%E0%B8%9A%E0%B8%B5%E0%B8%A2%E0%B8%9A%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B9%84%E0%B8%9F%E0%B9%84%E0%B8%99%E0%B8%95%E0%B9%8C%E0%B9%80%E0%B8%AD%E0%B8%A5%E0%B8%B4%E0%B9%80%E0%B8%A1%E0%B8%99%E0%B8%95%E0%B9%8C" title="ระเบียบวิธีไฟไนต์เอลิเมนต์ – Thai" lang="th" hreflang="th" data-title="ระเบียบวิธีไฟไนต์เอลิเมนต์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sonlu_elemanlar_y%C3%B6ntemi" title="Sonlu elemanlar yöntemi – Turkish" lang="tr" hreflang="tr" data-title="Sonlu elemanlar yöntemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D1%81%D0%BA%D1%96%D0%BD%D1%87%D0%B5%D0%BD%D0%BD%D0%B8%D1%85_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D1%96%D0%B2" title="Метод скінченних елементів – Ukrainian" lang="uk" hreflang="uk" data-title="Метод скінченних елементів" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_ph%C3%A1p_ph%E1%BA%A7n_t%E1%BB%AD_h%E1%BB%AFu_h%E1%BA%A1n" title="Phương pháp phần tử hữu hạn – Vietnamese" lang="vi" hreflang="vi" data-title="Phương pháp phần tử hữu hạn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E5%85%83%E5%88%86%E6%9E%90" title="有限元分析 – Cantonese" lang="yue" hreflang="yue" data-title="有限元分析" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E5%85%83%E7%B4%A0%E6%B3%95" title="有限元素法 – 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Finite element" redirects here. For the elements of a <a href="/wiki/Poset" class="mw-redirect" title="Poset">poset</a>, see <a href="/wiki/Compact_element" title="Compact element">compact element</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:FAE_visualization.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/FAE_visualization.jpg/250px-FAE_visualization.jpg" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/FAE_visualization.jpg/375px-FAE_visualization.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/FAE_visualization.jpg/500px-FAE_visualization.jpg 2x" data-file-width="700" data-file-height="525" /></a><figcaption>Visualization of how a car deforms in an asymmetrical crash using finite element analysis</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Scope</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;"> <div class="hlist"><ul><li><a href="/wiki/Natural_science" title="Natural science">Natural sciences</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Physics" title="Physics">Physics</a></li> <li><a href="/wiki/Chemistry" title="Chemistry">Chemistry</a></li> <li><br /><a href="/wiki/Biology" title="Biology">Biology</a></li> <li><a href="/wiki/Geology" title="Geology">Geology</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;"> <ul><li><a href="/wiki/Economics" title="Economics">Economics</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td> </tr></tbody></table> <hr /> <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;"> Classification</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> By variable type</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li> <li>Coupled / Decoupled</li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Features</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Operator</a></li></ul> </div> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size:85%;">(discrete analogue)</span></a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Solution</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Initial_condition" title="Initial condition">Initial conditions</a></li> <li><a href="/wiki/Boundary_value_problem" title="Boundary value problem">Boundary values</a> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><span class="nowrap"><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li>Inspection</li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li> <li><br /><a href="/wiki/Euler_method" title="Euler method">Euler</a></li> <li><a href="/wiki/Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size:85%;">(<a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li> <li><a class="mw-selflink selflink">Finite element</a> <ul><li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li> <li><a href="/wiki/Green%27s_function" title="Green's function">Green's function</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul> </div> <ul><li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> People</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations" title="Template:Differential equations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations" title="Template talk:Differential equations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations" title="Special:EditPage/Template:Differential equations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>finite element method</b> (<b>FEM</b>) is a popular method for numerically solving <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> arising in engineering and <a href="/wiki/Mathematical_models" class="mw-redirect" title="Mathematical models">mathematical modeling</a>. Typical problem areas of interest include the traditional fields of <a href="/wiki/Structural_analysis" title="Structural analysis">structural analysis</a>, <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, <a href="/wiki/Fluid_flow" class="mw-redirect" title="Fluid flow">fluid flow</a>, mass transport, and <a href="/wiki/Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potential</a>. Computers are usually used to perform the calculations required. With high-speed <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a>, better solutions can be achieved, and are often required to solve the largest and most complex problems. </p><p>The FEM is a general <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical method</a> for solving <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> in two or three space variables (i.e., some <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary value problems</a>). There are also studies about using FEM solve high-dimensional problems.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> To solve a problem, the FEM subdivides a large system into smaller, simpler parts called <b>finite elements</b>. This is achieved by a particular space <a href="/wiki/Discretization" title="Discretization">discretization</a> in the space dimensions, which is implemented by the construction of a <a href="/wiki/Types_of_mesh" title="Types of mesh">mesh</a> of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equations</a>. The method approximates the unknown function over the domain.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. </p><p>Studying or <a href="/wiki/Analysis" title="Analysis">analyzing</a> a phenomenon with FEM is often referred to as <b>finite element analysis</b> (<b>FEA</b>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=1" title="Edit section: Basic concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:608px;max-width:608px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Example_of_2D_mesh.png" class="mw-file-description"><img alt="Example of 2D mesh" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Example_of_2D_mesh.png/300px-Example_of_2D_mesh.png" decoding="async" width="300" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Example_of_2D_mesh.png/450px-Example_of_2D_mesh.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/80/Example_of_2D_mesh.png 2x" data-file-width="534" data-file-height="488" /></a></span></div><div class="thumbcaption">FEM <a href="/wiki/Polygon_mesh" title="Polygon mesh">mesh</a> created by an analyst before finding a solution to a <a href="/wiki/Magnetism" title="Magnetism">magnetic</a> problem using FEM software. Colors indicate that the analyst has set material properties for each zone, in this case, a <a href="/wiki/Electrical_conductor" title="Electrical conductor">conducting</a> wire coil in orange; a <a href="/wiki/Ferromagnetism" title="Ferromagnetism">ferromagnetic</a> component (perhaps <a href="/wiki/Iron" title="Iron">iron</a>) in light blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate the magnetic field for this setup without FEM software using <a href="/wiki/Closed-form_expression" title="Closed-form expression">equations alone</a>.</div></div><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:FEM_example_of_2D_solution.png" class="mw-file-description"><img alt="FEM_example_of_2D_solution" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/FEM_example_of_2D_solution.png/300px-FEM_example_of_2D_solution.png" decoding="async" width="300" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/FEM_example_of_2D_solution.png/450px-FEM_example_of_2D_solution.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7b/FEM_example_of_2D_solution.png 2x" data-file-width="534" data-file-height="488" /></a></span></div><div class="thumbcaption">FEM solution to the problem at left, involving a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylindrically</a> shaped <a href="/wiki/Magnetic_shielding" class="mw-redirect" title="Magnetic shielding">magnetic shield</a>. The <a href="/wiki/Ferromagnetism" title="Ferromagnetism">ferromagnetic</a> cylindrical part shields the area inside the cylinder by diverting the magnetic field <a href="/wiki/Electromagnet" title="Electromagnet">created</a> by the coil (rectangular area on the right). The color represents the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">amplitude</a> of the <a href="/wiki/Magnetic_field#Definitions,_units,_and_measurement" title="Magnetic field">magnetic flux density</a>, as indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is low amplitude (dark blue, with widely spaced lines of magnetic flux), which suggests that the shield is performing as it was designed to.</div></div></div></div></div> <p>The subdivision of a whole domain into simpler parts has several advantages:<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Accurate representation of complex geometry</li> <li>Inclusion of dissimilar material properties</li> <li>Easy representation of the total solution</li> <li>Capture of local effects.</li></ul> <p>Typical work out of the method involves: </p> <ol><li>dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem</li> <li>systematically recombining all sets of element equations into a global system of equations for the final calculation.</li></ol> <p>The global system of equations has known solution techniques and can be calculated from the <a href="/wiki/Initial_value" class="mw-redirect" title="Initial value">initial values</a> of the original problem to obtain a numerical answer. </p><p>In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of <a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a>. The process, in mathematical language, is to construct an integral of the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> of the residual and the <a href="/wiki/Weight_function" title="Weight function">weight functions</a> and set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with </p> <ul><li>a set of <a href="/wiki/Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a> for <a href="/wiki/Steady_state" title="Steady state">steady state</a> problems,</li> <li>a set of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> for <a href="/wiki/Transient_state" title="Transient state">transient</a> problems.</li></ul> <p>These equation sets are element equations. They are <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a> methods. In contrast, <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> sets that occur in the transient problems are solved by numerical integration using standard techniques such as <a href="/wiki/Euler%27s_method" class="mw-redirect" title="Euler's method">Euler's method</a> or the <a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge-Kutta method</a>. </p><p>In step (2) above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate <a href="/wiki/Transformation_matrix" title="Transformation matrix">orientation adjustments</a> as applied in relation to the reference <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. The process is often carried out by FEM software using <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinate</a> data generated from the subdomains. </p><p>The practical application of FEM is known as <i>finite element analysis</i> (FEA). FEA as applied in <a href="/wiki/Engineering" title="Engineering">engineering</a>, is a computational tool for performing <a href="/wiki/Engineering_analysis" title="Engineering analysis">engineering analysis</a>. It includes the use of <a href="/wiki/Mesh_generation" title="Mesh generation">mesh generation</a> techniques for dividing a <a href="/wiki/Complex_system" title="Complex system">complex problem</a> into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying <a href="/wiki/Physics" title="Physics">physics</a> such as the <a href="/wiki/Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">Euler–Bernoulli beam equation</a>, the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>, or the <a href="/wiki/Navier-Stokes_equations" class="mw-redirect" title="Navier-Stokes equations">Navier-Stokes equations</a> expressed in either PDE or <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a>, while the divided small elements of the complex problem represent different areas in the physical system. </p><p>FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines) when the domain changes (as during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2021)">citation needed</span></a></i>]</sup> For example, in a frontal crash simulation, it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing the cost of the simulation). Another example would be in <a href="/wiki/Numerical_weather_prediction" title="Numerical weather prediction">numerical weather prediction</a>, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as <a href="/wiki/Tropical_cyclone" title="Tropical cyclone">tropical cyclones</a> in the atmosphere, or <a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">eddies</a> in the ocean) rather than relatively calm areas. </p><p>A clear, detailed, and practical presentation of this approach can be found in the textbook <i>The Finite Element Method for Engineers</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a> and <a href="/wiki/Structural_analysis" title="Structural analysis">structural analysis</a> problems in <a href="/wiki/Civil_engineering" title="Civil engineering">civil</a> and <a href="/wiki/Aeronautical_engineering" class="mw-redirect" title="Aeronautical engineering">aeronautical engineering</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Its development can be traced back to work by <a href="/wiki/Alexander_Hrennikoff" title="Alexander Hrennikoff">Alexander Hrennikoff</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Richard_Courant" title="Richard Courant">Richard Courant</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> in the early 1940s. Another pioneer was <a href="/wiki/Ioannis_Argyris" class="mw-redirect" title="Ioannis Argyris">Ioannis Argyris</a>. In the USSR, the introduction of the practical application of the method is usually connected with the name of <a href="/w/index.php?title=Leonard_Oganesyan&action=edit&redlink=1" class="new" title="Leonard Oganesyan (page does not exist)">Leonard Oganesyan</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It was also independently rediscovered in China by <a href="/wiki/Feng_Kang" title="Feng Kang">Feng Kang</a> in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the <i>finite difference method based on variation principle</i>. Although the approaches used by these pioneers are different, they share one essential characteristic: <a href="/wiki/Polygon_mesh" title="Polygon mesh">mesh</a> <a href="/wiki/Discretization" title="Discretization">discretization</a> of a continuous domain into a set of discrete sub-domains, usually called elements. </p><p>Hrennikoff's work discretizes the domain by using a <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> analogy, while Courant's approach divides the domain into finite triangular subregions to solve <a href="/wiki/Partial_differential_equation#Linear_equations_of_second_order" title="Partial differential equation">second order</a> <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equations</a> that arise from the problem of <a href="/wiki/Torsion_(mechanics)" title="Torsion (mechanics)">torsion</a> of a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a>. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by <a href="/wiki/John_William_Strutt,_3rd_Baron_Rayleigh" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a>, <a href="/wiki/Walther_Ritz" title="Walther Ritz">Walther Ritz</a>, and <a href="/wiki/Boris_Galerkin" title="Boris Galerkin">Boris Galerkin</a>. </p><p>The finite element method obtained its real impetus in the 1960s and 1970s by the developments of <a href="/wiki/John_Argyris" title="John Argyris">J. H. Argyris</a> with co-workers at the <a href="/wiki/University_of_Stuttgart" title="University of Stuttgart">University of Stuttgart</a>, <a href="/wiki/Ray_W._Clough" class="mw-redirect" title="Ray W. Clough">R. W. Clough</a> with co-workers at <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">UC Berkeley</a>, <a href="/wiki/Olgierd_Zienkiewicz" title="Olgierd Zienkiewicz">O. C. Zienkiewicz</a> with co-workers <a href="/wiki/Ernest_Hinton" title="Ernest Hinton">Ernest Hinton</a>, <a href="/wiki/Bruce_Irons_(engineer)" title="Bruce Irons (engineer)">Bruce Irons</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and others at <a href="/wiki/Swansea_University" title="Swansea University">Swansea University</a>, <a href="/wiki/Philippe_G._Ciarlet" title="Philippe G. Ciarlet">Philippe G. Ciarlet</a> at the University of <a href="/wiki/Pierre-and-Marie-Curie_University" class="mw-redirect" title="Pierre-and-Marie-Curie University">Paris 6</a> and <a href="/wiki/Richard_H._Gallagher" title="Richard H. Gallagher">Richard Gallagher</a> with co-workers at <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>. Further impetus was provided in these years by available open-source finite element programs. NASA sponsored the original version of <a href="/wiki/NASTRAN" class="mw-redirect" title="NASTRAN">NASTRAN</a>. UC Berkeley made the finite element programs SAP IV<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and later <a href="/wiki/OpenSees" title="OpenSees">OpenSees</a> widely available. In Norway, the ship classification society Det Norske Veritas (now <a href="/wiki/DNV_GL" class="mw-redirect" title="DNV GL">DNV GL</a>) developed <a href="/wiki/SESAM_(FEM)" class="mw-redirect" title="SESAM (FEM)">Sesam</a> in 1969 for use in the analysis of ships.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by <a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a> and <a href="/wiki/George_Fix" title="George Fix">George Fix</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The method has since been generalized for the <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical modeling</a> of physical systems in a wide variety of <a href="/wiki/Engineering" title="Engineering">engineering</a> disciplines, e.g., <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, and <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>.<sup id="cite_ref-ZienkiewiczTaylor2013_13-0" class="reference"><a href="#cite_note-ZienkiewiczTaylor2013-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Technical_discussion">Technical discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=3" title="Edit section: Technical discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_structure_of_finite_element_methods">The structure of finite element methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=4" title="Edit section: The structure of finite element methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A finite element method is characterized by a <a href="/wiki/Calculus_of_variations" title="Calculus of variations">variational formulation</a>, a discretization strategy, one or more solution algorithms, and post-processing procedures. </p><p>Examples of the variational formulation are the <a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a>, the discontinuous Galerkin method, mixed methods, etc. </p><p>A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions), and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, <a href="/wiki/P-FEM" title="P-FEM">p-version</a>, <a href="/wiki/Hp-FEM" title="Hp-FEM">hp-version</a>, <a href="/wiki/Extended_finite_element_method" title="Extended finite element method">x-FEM</a>, <a href="/wiki/Isogeometric_analysis" title="Isogeometric analysis">isogeometric analysis</a>, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class. </p><p>Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices. </p><p>Post-processing procedures are designed to extract the data of interest from a finite element solution. To meet the requirements of solution verification, postprocessors need to provide for <i>a posteriori</i> error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. Some very efficient postprocessors provide for the realization of <a href="/wiki/Superconvergence" title="Superconvergence">superconvergence</a>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Illustrative_problems_P1_and_P2">Illustrative problems P1 and P2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=5" title="Edit section: Illustrative problems P1 and P2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following two problems demonstrate the finite element method. </p><p>P1 is a one-dimensional problem <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext> P1 </mtext> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>u</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> in </mtext> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6154fa70d676595f41422ad30c52cd64c1b59b35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.376ex; height:6.176ex;" alt="{\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is given, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> is an unknown function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75057c7f0c536c8aa2f1df42cfddcfd96985c4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle u''}"></span> is the second derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p><p>P2 is a two-dimensional problem (<a href="/wiki/Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>P2 </mtext> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> in </mtext> </mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> on </mtext> </mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2339a9e3170d7082303019b6f8d20c31a2f64e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.181ex; height:6.176ex;" alt="{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is a connected open region in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> plane whose boundary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16feddaad462c2a1c9efdaeee062a0484a023fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }"></span> is nice (e.g., a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> or a <a href="/wiki/Polygon" title="Polygon">polygon</a>), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c972226aadbf63510b7610318e457526e685b681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.442ex; height:2.009ex;" alt="{\displaystyle u_{xx}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{yy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{yy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5547e142f37adaf3dc62a0a9ac8e312bbf18e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.196ex; height:2.343ex;" alt="{\displaystyle u_{yy}}"></span> denote the second derivatives with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, respectively. </p><p>The problem P1 can be solved directly by computing <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a>. However, this method of solving the <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary value problem</a> (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u+V''=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>+</mo> <msup> <mi>V</mi> <mo>″</mo> </msup> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u+V''=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b54f505b218da7280f8d50896868885e7421f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.601ex; height:2.843ex;" alt="{\displaystyle u+V''=f}"></span>. For this reason, we will develop the finite element method for P1 and outline its generalization to P2. </p><p>Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM. </p> <ul><li>In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper.</li> <li>The second step is discretization, where the weak form is discretized in a finite-dimensional space.</li></ul> <p>After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a <a href="/wiki/Computer" title="Computer">computer</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Weak_formulation">Weak formulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=6" title="Edit section: Weak formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first step is to convert P1 and P2 into their equivalent <a href="/wiki/Weak_formulation" title="Weak formulation">weak formulations</a>. </p> <div class="mw-heading mw-heading4"><h4 id="The_weak_form_of_P1">The weak form of P1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=7" title="Edit section: The weak form of P1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> solves P1, then for any smooth function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> that satisfies the displacement boundary conditions, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"></span>, we have </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>u</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c5419715b7317cd7ea0475c599d37781bbf0ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.978ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>Conversely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(0)=u(1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(0)=u(1)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a34dfe771c1ceeeddd0fd34558f2e0777de5d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.962ex; height:2.843ex;" alt="{\displaystyle u(0)=u(1)=0}"></span> satisfies (1) for every smooth function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b371a381e15c71d8fc4ec43cf14b156f02a0d35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.267ex; height:2.843ex;" alt="{\displaystyle v(x)}"></span> then one may show that this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> will solve P1. The proof is easier for twice continuously differentiable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> (<a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>) but may be proved in a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributional</a> sense as well. </p><p>We define a new operator or map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857f5ddba44084b873ec4c89532a69db8350e0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \phi (u,v)}"></span> by using <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a> on the right-hand-side of (1): </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&=\int _{0}^{1}u''(x)v(x)\,dx\\&=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>u</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≡</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&=\int _{0}^{1}u''(x)v(x)\,dx\\&=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aca8e65b05e6fd22bf65e79f8abb32de0537a59e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.74ex; margin-bottom: -0.265ex; width:50.75ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&=\int _{0}^{1}u''(x)v(x)\,dx\\&=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>where we have used the assumption that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(0)=v(1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(0)=v(1)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918328090764fcd5813a5c42ee1fd5c06565e4cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.558ex; height:2.843ex;" alt="{\displaystyle v(0)=v(1)=0}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="The_weak_form_of_P2">The weak form of P2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=8" title="Edit section: The weak form of P2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we integrate by parts using a form of <a href="/wiki/Green%27s_identities" title="Green's identities">Green's identities</a>, we see that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> solves P2, then we may define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857f5ddba44084b873ec4c89532a69db8350e0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \phi (u,v)}"></span> for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>f</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>≡</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3093300ada5159d6a58627bf9a1d9e27eb42f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.531ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></span> denotes the <a href="/wiki/Gradient" title="Gradient">gradient</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> denotes the <a href="/wiki/Dot_product" title="Dot product">dot product</a> in the two-dimensional plane. Once more <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\!\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53894ae6e1efd0e272291abea516a222ace8a4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \,\!\phi }"></span> can be turned into an inner product on a suitable space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830c3c449349317a46b77590a546b7af4a62dc08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(\Omega )}"></span> of once differentiable functions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> that are zero on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16feddaad462c2a1c9efdaeee062a0484a023fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }"></span>. We have also assumed that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in H_{0}^{1}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in H_{0}^{1}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b528325e0f9461d9c2710717a5cde8f2c7116f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.613ex; height:3.176ex;" alt="{\displaystyle v\in H_{0}^{1}(\Omega )}"></span> (see <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a>). The existence and uniqueness of the solution can also be shown. </p> <div class="mw-heading mw-heading4"><h4 id="A_proof_outline_of_the_existence_and_uniqueness_of_the_solution">A proof outline of the existence and uniqueness of the solution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=9" title="Edit section: A proof outline of the existence and uniqueness of the solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can loosely think of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d3d5f81eae348b97a89a1b17d6c84e01544366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}"></span> to be the <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> functions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> that are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"></span> (see <a href="/wiki/Sobolev_spaces" class="mw-redirect" title="Sobolev spaces">Sobolev spaces</a>). Such functions are (weakly) once differentiable, and it turns out that the symmetric <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear map</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \!\,\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \!\,\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1315e3582db5f2a61f20766851bad6f0860e5069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.387ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \!\,\phi }"></span> then defines an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> which turns <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d3d5f81eae348b97a89a1b17d6c84e01544366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}"></span> into a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> (a detailed proof is nontrivial). On the other hand, the left-hand-side <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}f(x)v(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}f(x)v(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca47c7a1b9cf10efac7b342c6dccad88f97a16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.135ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}f(x)v(x)dx}"></span> is also an inner product, this time on the <a href="/wiki/Lp_space" title="Lp space">Lp space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc63d7b243a51a699bcb6d6cf30b6ca2a4a65a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.805ex; height:3.176ex;" alt="{\displaystyle L^{2}(0,1)}"></span>. An application of the <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a> for Hilbert spaces shows that there is a unique <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> solving (2) and, therefore, P1. This solution is a-priori only a member of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d3d5f81eae348b97a89a1b17d6c84e01544366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}"></span>, but using <a href="/wiki/Elliptic_operator" title="Elliptic operator">elliptic</a> regularity, will be smooth if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is. </p> <div class="mw-heading mw-heading2"><h2 id="Discretization">Discretization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=10" title="Edit section: Discretization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finite_element_method_1D_illustration1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Finite_element_method_1D_illustration1.png/220px-Finite_element_method_1D_illustration1.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Finite_element_method_1D_illustration1.png/330px-Finite_element_method_1D_illustration1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Finite_element_method_1D_illustration1.png/440px-Finite_element_method_1D_illustration1.png 2x" data-file-width="1002" data-file-height="651" /></a><figcaption>A function in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513d4d6147067441ca45942e5577034393045d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.805ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1},}"></span> with zero values at the endpoints (blue) and a piecewise linear approximation (red)</figcaption></figure> <p>P1 and P2 are ready to be discretized, which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem: </p> <dl><dd>Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in H_{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in H_{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13251cb612942c56abf6aa865b546cf5204f0932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.328ex; height:3.176ex;" alt="{\displaystyle u\in H_{0}^{1}}"></span> such that</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb0a124f7f851f7ffda822cafc67b8f8f3dccc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.677ex; height:5.676ex;" alt="{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}"></span></dd></dl> <p>with a finite-dimensional version: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap">Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dd20088dea1139b38b3c04053ccf508bbed8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}"></span> such that<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c819c7a75457354318c1ce1f3a94b92f42f9b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.307ex; height:5.676ex;" alt="{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is a finite-dimensional <a href="/wiki/Linear_subspace" title="Linear subspace">subspace</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9c9e1684f33b9116efe894347370f5bfc0c8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}"></span>. There are many possible choices for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (one possibility leads to the <a href="/wiki/Spectral_method" title="Spectral method">spectral method</a>). However, we take <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> as a space of piecewise polynomial functions for the finite element method. </p> <div class="mw-heading mw-heading3"><h3 id="For_problem_P1">For problem P1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=11" title="Edit section: For problem P1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We take the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>, choose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7730812aae7cfa944b21b6e06c10f611c91433d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.604ex; height:2.509ex;" alt="{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}"></span> and we define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mo>:</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is continuous, </mtext> </mrow> <mi>v</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> is linear for </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>, and </mtext> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625e1faf3479f6031890c60c91e234b603830482" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:93.084ex; height:3.343ex;" alt="{\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}"></span> </p><p>where we define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d18a96da37e1748deeb8d4c590dd4ad6629efef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{0}=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094e2edc998bc87ff0e124c456fb9cf757f92141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.91ex; height:2.509ex;" alt="{\displaystyle x_{n+1}=1}"></span>. Observe that functions in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> are not differentiable according to the elementary definition of calculus. Indeed, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span> then the derivative is typically not defined at any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e676498d1003087514f2b5ad431f1e73945c565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.847ex; height:2.009ex;" alt="{\displaystyle x=x_{k}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02703686f808b37fedb436806fa72ca3522e22de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,n}"></span>. However, the derivative exists at every other value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, and one can use this derivative for <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Piecewise_linear_function2D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Piecewise_linear_function2D.svg/220px-Piecewise_linear_function2D.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Piecewise_linear_function2D.svg/330px-Piecewise_linear_function2D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Piecewise_linear_function2D.svg/440px-Piecewise_linear_function2D.svg.png 2x" data-file-width="443" data-file-height="443" /></a><figcaption>A piecewise linear function in two dimensions</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="For_problem_P2">For problem P2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=12" title="Edit section: For problem P2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We need <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> to be a set of functions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. In the figure on the right, we have illustrated a <a href="/wiki/Polygon_triangulation" title="Polygon triangulation">triangulation</a> of a 15-sided <a href="/wiki/Polygon" title="Polygon">polygonal</a> region <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> in the plane (below), and a <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear function</a> (above, in color) of this polygon which is linear on each triangle of the triangulation; the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> would consist of functions that are linear on each triangle of the chosen triangulation. </p><p>One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbddb7a5cca6170575e4e73e769fbb434c2a3d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}"></span> which one takes to be very small. This parameter will be related to the largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> must also change with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. For this reason, one often reads <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652f5cdfa49da86f90fa98f1ab5c47a3384f1464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle V_{h}}"></span> instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> in the literature. Since we do not perform such an analysis, we will not use this notation. </p> <div class="mw-heading mw-heading3"><h3 id="Choosing_a_basis">Choosing a basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=13" title="Edit section: Choosing a basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader" style="text-align:center">Interpolation of a <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Linear_interpolation_of_J0_(basis_set).svg" class="mw-file-description"><img alt="Sixteen triangular basis functions used to reconstruct J0" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Linear_interpolation_of_J0_%28basis_set%29.svg/200px-Linear_interpolation_of_J0_%28basis_set%29.svg.png" decoding="async" width="200" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Linear_interpolation_of_J0_%28basis_set%29.svg/300px-Linear_interpolation_of_J0_%28basis_set%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Linear_interpolation_of_J0_%28basis_set%29.svg/400px-Linear_interpolation_of_J0_%28basis_set%29.svg.png 2x" data-file-width="719" data-file-height="444" /></a></span></div><div class="thumbcaption text-align-left">16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function <i>J</i><sub><i>0</i></sub> (black)</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Linear_interpolation_of_J1_(basis_set).svg" class="mw-file-description"><img alt="Summation of basis functions" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Linear_interpolation_of_J1_%28basis_set%29.svg/200px-Linear_interpolation_of_J1_%28basis_set%29.svg.png" decoding="async" width="200" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Linear_interpolation_of_J1_%28basis_set%29.svg/300px-Linear_interpolation_of_J1_%28basis_set%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Linear_interpolation_of_J1_%28basis_set%29.svg/400px-Linear_interpolation_of_J1_%28basis_set%29.svg.png 2x" data-file-width="719" data-file-height="444" /></a></span></div><div class="thumbcaption text-align-left">The linear combination of basis functions (yellow) reproduces <i>J</i><sub><i>0</i></sub> (black) to any desired accuracy.</div></div></div></div></div> <p>To complete the discretization, we must select a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. In the one-dimensional case, for each control point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> we will choose the piecewise linear function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> whose value is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> and zero at every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},\;j\neq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>≠</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j},\;j\neq k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f729084d28d94a3aa9f08ed2e4cdcf7e4f84b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.186ex; height:2.843ex;" alt="{\displaystyle x_{j},\;j\neq k}"></span>, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in [x_{k},x_{k+1}],\\0&{\text{ otherwise}},\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>−</mo> <mi>x</mi> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> otherwise</mtext> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in [x_{k},x_{k+1}],\\0&{\text{ otherwise}},\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb00fac29c2b4782376483b056cd062a8cf816ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:38.003ex; height:11.176ex;" alt="{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in [x_{k},x_{k+1}],\\0&{\text{ otherwise}},\end{cases}}}"></span> </p><p>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cea3d30a653b96d33958125a11c37add9d66a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\dots ,n}"></span>; this basis is a shifted and scaled <a href="/wiki/Tent_function" class="mw-redirect" title="Tent function">tent function</a>. For the two-dimensional case, we choose again one basis function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> per vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> of the triangulation of the planar region <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> is the unique function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> whose value is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> and zero at every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},\;j\neq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>≠</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j},\;j\neq k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f729084d28d94a3aa9f08ed2e4cdcf7e4f84b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.186ex; height:2.843ex;" alt="{\displaystyle x_{j},\;j\neq k}"></span>. </p><p>Depending on the author, the word "element" in the "finite element method" refers to the domain's triangles, the piecewise linear basis function, or both. So, for instance, an author interested in curved domains might replace the triangles with curved primitives and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle). </p><p>Examples of methods that use higher degree piecewise polynomial basis functions are the <a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a> and <a href="/wiki/Spectral_element_method" title="Spectral element method">spectral FEM</a>. </p><p>More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques; the most popular are: </p> <ul><li>moving nodes (r-adaptivity)</li> <li>refining (and unrefined) elements (h-adaptivity)</li> <li>changing order of base functions (p-adaptivity)</li> <li>combinations of the above (<a href="/wiki/Hp-FEM" title="Hp-FEM">hp-adaptivity</a>).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Small_support_of_the_basis">Small support of the basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=14" title="Edit section: Small support of the basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finite_element_triangulation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Finite_element_triangulation.svg/220px-Finite_element_triangulation.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Finite_element_triangulation.svg/330px-Finite_element_triangulation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Finite_element_triangulation.svg/440px-Finite_element_triangulation.svg.png 2x" data-file-width="815" data-file-height="815" /></a><figcaption>Solving the two-dimensional problem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{xx}+u_{yy}=-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xx}+u_{yy}=-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7d65e3fa0569101ce243bbdaf5297498d7f335" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.548ex; height:2.843ex;" alt="{\displaystyle u_{xx}+u_{yy}=-4}"></span> in the disk centered at the origin and radius 1, with zero boundary conditions.<br />(a) The triangulation.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finite_element_sparse_matrix.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Finite_element_sparse_matrix.png/220px-Finite_element_sparse_matrix.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Finite_element_sparse_matrix.png/330px-Finite_element_sparse_matrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Finite_element_sparse_matrix.png/440px-Finite_element_sparse_matrix.png 2x" data-file-width="816" data-file-height="816" /></a><figcaption>(b) The <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse matrix</a> <i>L</i> of the discretized linear system</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finite_element_solution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Finite_element_solution.svg/220px-Finite_element_solution.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Finite_element_solution.svg/330px-Finite_element_solution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Finite_element_solution.svg/440px-Finite_element_solution.svg.png 2x" data-file-width="1235" data-file-height="863" /></a><figcaption>(c) The computed solution, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x,y)=1-x^{2}-y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,y)=1-x^{2}-y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f759b0e46d5413c2fa31120bfe970450ca6175e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.198ex; height:3.176ex;" alt="{\displaystyle u(x,y)=1-x^{2}-y^{2}}"></span></figcaption></figure> <p>The primary advantage of this choice of basis is that the inner products <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a5152e981da032ae13d1ecd73f3903b79f4904" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.287ex; height:6.176ex;" alt="{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e299dd96b556aab2d84f6470f902ae9216194019" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.673ex; height:6.176ex;" alt="{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}"></span> will be zero for almost all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j,k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23e18a251a10a993e66d41e8dbcaf858ba4fa5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle j,k}"></span>. (The matrix containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a4c45ed6c6dc82ecae85127175860afb2f67bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.097ex; height:3.009ex;" alt="{\displaystyle \langle v_{j},v_{k}\rangle }"></span> in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (j,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (j,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d001cfb64ce123bb1a5ddf9047cd949d7753145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle (j,k)}"></span> location is known as the <a href="/wiki/Gramian_matrix" class="mw-redirect" title="Gramian matrix">Gramian matrix</a>.) In the one dimensional case, the <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> is the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{k-1},x_{k+1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{k-1},x_{k+1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ab905e7068db95aeddfd9f0fa55af0e0363843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.365ex; height:2.843ex;" alt="{\displaystyle [x_{k-1},x_{k+1}]}"></span>. Hence, the integrands of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a4c45ed6c6dc82ecae85127175860afb2f67bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.097ex; height:3.009ex;" alt="{\displaystyle \langle v_{j},v_{k}\rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (v_{j},v_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (v_{j},v_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a01aec768783146ac1c1b74d1aa362ca0065af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.482ex; height:3.009ex;" alt="{\displaystyle \phi (v_{j},v_{k})}"></span> are identically zero whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |j-k|>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |j-k|>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fc83292a5fed7bcc2e8f170da5ec530ef32aa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.564ex; height:2.843ex;" alt="{\displaystyle |j-k|>1}"></span>. </p><p>Similarly, in the planar case, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> do not share an edge of the triangulation, then the integrals <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e28c019587634c339f5f226018ad7d2092bbb9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.046ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942130c0c388bdf4a73c7b153d405e97bb734dd6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.596ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}"></span> are both zero. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_form_of_the_problem">Matrix form of the problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=15" title="Edit section: Matrix form of the problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d048e6042ecb9920d5d78693f9cd0e162cc4567e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.083ex; height:6.843ex;" alt="{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8755493ff0ad90921dcd691ea6bbf55810f96aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.841ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}"></span> then problem (3), taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)=v_{j}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(x)=v_{j}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffac80797632924fc24db91fa08fe38ed82f3ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.541ex; height:3.009ex;" alt="{\displaystyle v(x)=v_{j}(x)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9393eaa189f1fb2c747b687b7b8d67640d5f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}"></span>, becomes </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∫</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b7db3f81feaa3057a676e7d964092f28dffc7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.674ex; height:6.843ex;" alt="{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78fbe66f1d761a4c3189e16cfb5459335c8b20c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.465ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n.}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <p>If we denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"></span> the column vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{1},\dots ,u_{n})^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{1},\dots ,u_{n})^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3a99d74915a3d8c937df712e359fc849c7fb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.746ex; height:3.009ex;" alt="{\displaystyle (u_{1},\dots ,u_{n})^{t}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{1},\dots ,f_{n})^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{1},\dots ,f_{n})^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf014eef21cf396342cb9bf18ee343c5ac65552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.365ex; height:3.009ex;" alt="{\displaystyle (f_{1},\dots ,f_{n})^{t}}"></span>, and if we let <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=(L_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=(L_{ij})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99c776f96b08197ed290d92c8402945cdb5ea59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.551ex; height:3.009ex;" alt="{\displaystyle L=(L_{ij})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=(M_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=(M_{ij})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d431bad9e88eb84a968d6865755a82d2416189a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.081ex; height:3.009ex;" alt="{\displaystyle M=(M_{ij})}"></span> be matrices whose entries are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}=\phi (v_{i},v_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ij}=\phi (v_{i},v_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2320bcda62e8747a42c7c13c8c615fbab62c21e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.352ex; height:3.009ex;" alt="{\displaystyle L_{ij}=\phi (v_{i},v_{j})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{ij}=\int v_{i}v_{j}dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}=\int v_{i}v_{j}dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48b83849a78d485ee5fbfb230bba36faa80b682" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.921ex; height:5.676ex;" alt="{\displaystyle M_{ij}=\int v_{i}v_{j}dx}"></span> then we may rephrase (4) as </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -L\mathbf {u} =M\mathbf {f} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -L\mathbf {u} =M\mathbf {f} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1964d1f50831a532d92becc41de7a0644055ad55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.117ex; height:2.343ex;" alt="{\displaystyle -L\mathbf {u} =M\mathbf {f} .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <p>It is not necessary to assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8755493ff0ad90921dcd691ea6bbf55810f96aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.841ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}"></span>. For a general function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>, problem (3) with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)=v_{j}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(x)=v_{j}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffac80797632924fc24db91fa08fe38ed82f3ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.541ex; height:3.009ex;" alt="{\displaystyle v(x)=v_{j}(x)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9393eaa189f1fb2c747b687b7b8d67640d5f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}"></span> becomes actually simpler, since no matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is used, </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -L\mathbf {u} =\mathbf {b} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -L\mathbf {u} =\mathbf {b} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4f7291174e2a20eb95513c8b9dfb6b08e4667c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.107ex; height:2.509ex;" alt="{\displaystyle -L\mathbf {u} =\mathbf {b} ,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9133ba3635dd43f495d260c45bde8bd61a60090c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.665ex; height:3.009ex;" alt="{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=\int fv_{j}dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{j}=\int fv_{j}dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e3e00504a55ef0b934e99e6f7516cd356a44b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.448ex; height:5.676ex;" alt="{\displaystyle b_{j}=\int fv_{j}dx}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9393eaa189f1fb2c747b687b7b8d67640d5f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}"></span>. </p><p>As we have discussed before, most of the entries of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> are zero because the basis functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> have small support. So we now have to solve a linear system in the unknown <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span> where most of the entries of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, which we need to invert, are zero. </p><p>Such matrices are known as <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse matrices</a>, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is symmetric and positive definite, so a technique such as the <a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a> is favored. For problems that are not too large, sparse <a href="/wiki/LU_decomposition" title="LU decomposition">LU decompositions</a> and <a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decompositions</a> still work well. For instance, <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>'s backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices. </p><p>The matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is usually referred to as the <a href="/wiki/Stiffness_matrix" title="Stiffness matrix">stiffness matrix</a>, while the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is dubbed the <a href="/wiki/Mass_matrix" title="Mass matrix">mass matrix</a>. </p> <div class="mw-heading mw-heading3"><h3 id="General_form_of_the_finite_element_method">General form of the finite element method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=16" title="Edit section: General form of the finite element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, the finite element method is characterized by the following process. </p> <ul><li>One chooses a grid for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.</li> <li>Then, one chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions.</li></ul> <p>Separate consideration is the smoothness of the basis functions. For second-order <a href="/wiki/Elliptic_boundary_value_problem" title="Elliptic boundary value problem">elliptic boundary value problems</a>, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higher-order partial differential equations, one must use smoother basis functions. For instance, for a fourth-order problem such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{xxxx}+u_{yyyy}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xxxx}+u_{yyyy}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d1831f74bd8583801790de444e619ec2f17611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.37ex; height:2.843ex;" alt="{\displaystyle u_{xxxx}+u_{yyyy}=f}"></span>, one may use piecewise quadratic basis functions that are <a href="/wiki/Smooth_function#Order_of_continuity" class="mw-redirect" title="Smooth function"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}"></span></a>. </p><p>Another consideration is the relation of the finite-dimensional space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> to its infinite-dimensional counterpart in the examples above <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9c9e1684f33b9116efe894347370f5bfc0c8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}"></span>. A <a href="/w/index.php?title=Conforming_element_method&action=edit&redlink=1" class="new" title="Conforming element method (page does not exist)">conforming element method</a> is one in which space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a <a href="/w/index.php?title=Nonconforming_element_method&action=edit&redlink=1" class="new" title="Nonconforming element method (page does not exist)">nonconforming element method</a>, an example of which is the space of piecewise linear functions over the mesh, which are continuous at each edge midpoint. Since these functions are generally discontinuous along the edges, this finite-dimensional space is not a subspace of the original <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9c9e1684f33b9116efe894347370f5bfc0c8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}"></span>. </p><p>Typically, one has an algorithm for subdividing a given mesh. If the primary method for increasing precision is to subdivide the mesh, one has an <i>h</i>-method (<i>h</i> is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is bounded above by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ch^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ch^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08cd8eb70811c4b4b41fd9e43382084ecd068c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.164ex; height:2.343ex;" alt="{\displaystyle Ch^{p}}"></span>, for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbfe849f5960e5d8dddf496797342677b0b2fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.189ex; height:2.176ex;" alt="{\displaystyle C<\infty }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span>, then one has an order <i>p</i> method. Under specific hypotheses (for instance, if the domain is convex), a piecewise polynomial of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> method will have an error of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=d+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=d+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fe11ba16f2cfa26d675787bda9cc3bcf4fc0e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.576ex; height:2.509ex;" alt="{\displaystyle p=d+1}"></span>. </p><p>If instead of making <i>h</i> smaller, one increases the degree of the polynomials used in the basis function, one has a <i>p</i>-method. If one combines these two refinement types, one obtains an <i>hp</i>-method (<a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a>). In the hp-FEM, the polynomial degrees can vary from element to element. High-order methods with large uniform <i>p</i> are called spectral finite element methods (<a href="/wiki/Spectral_element_method" title="Spectral element method">SFEM</a>). These are not to be confused with <a href="/wiki/Spectral_method" title="Spectral method">spectral methods</a>. </p><p>For vector partial differential equations, the basis functions may take values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Various_types_of_finite_element_methods">Various types of finite element methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=17" title="Edit section: Various types of finite element methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="AEM">AEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=18" title="Edit section: AEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Applied Element Method or AEM combines features of both FEM and <a href="/wiki/Discrete_element_method" title="Discrete element method">Discrete element method</a> or (DEM). </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Applied_element_method" title="Applied element method">Applied element method</a></div> <div class="mw-heading mw-heading3"><h3 id="A-FEM">A-FEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=19" title="Edit section: A-FEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Yang and Lui introduced the Augmented-Finite Element Method, whose goal was to model the weak and strong discontinuities without needing extra DoFs, as PuM stated. </p> <div class="mw-heading mw-heading3"><h3 id="CutFEM">CutFEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=20" title="Edit section: CutFEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Cut Finite Element Approach was developed in 2014.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The approach is "to make the discretization as independent as possible of the geometric description and minimize the complexity of mesh generation, while retaining the accuracy and robustness of a standard finite element method."<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_finite_element_method">Generalized finite element method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=21" title="Edit section: Generalized finite element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a <a href="/wiki/Partition_of_unity" title="Partition of unity">partition of unity</a> is used to “bond” these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mixed_finite_element_method">Mixed finite element method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=22" title="Edit section: Mixed finite element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mixed_finite_element_method" title="Mixed finite element method">Mixed finite element method</a></div> <p>The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem. </p> <div class="mw-heading mw-heading3"><h3 id="Variable_–_polynomial"><span id="Variable_.E2.80.93_polynomial"></span>Variable – polynomial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=23" title="Edit section: Variable – polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a> combines adaptively elements with variable size <i>h</i> and polynomial degree <i>p</i> to achieve exceptionally fast, exponential convergence rates.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="hpk-FEM">hpk-FEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=24" title="Edit section: hpk-FEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/w/index.php?title=Hpk-FEM&action=edit&redlink=1" class="new" title="Hpk-FEM (page does not exist)">hpk-FEM</a> combines adaptively elements with variable size <i>h</i>, polynomial degree of the local approximations <i>p</i>, and global differentiability of the local approximations (<i>k</i>-1) to achieve the best convergence rates. </p> <div class="mw-heading mw-heading3"><h3 id="XFEM">XFEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=25" title="Edit section: XFEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Extended_finite_element_method" title="Extended finite element method">Extended finite element method</a></div> <p>The <a href="/wiki/Extended_finite_element_method" title="Extended finite element method">extended finite element method</a> (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods at the cost of restricting the discontinuities to mesh edges. </p><p>Several research codes implement this technique to various degrees: </p> <ol><li>GetFEM++</li> <li>xfem++</li> <li>openxfem++</li></ol> <p>XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.). </p> <div class="mw-heading mw-heading3"><h3 id="Scaled_boundary_finite_element_method_(SBFEM)"><span id="Scaled_boundary_finite_element_method_.28SBFEM.29"></span>Scaled boundary finite element method (SBFEM)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=26" title="Edit section: Scaled boundary finite element method (SBFEM)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997).<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semi-analytical fundamental-solutionless method combining the advantages of finite element formulations and procedures and boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required. </p> <div class="mw-heading mw-heading3"><h3 id="S-FEM">S-FEM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=27" title="Edit section: S-FEM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed finite element method</a></div> <p>The S-FEM, Smoothed Finite Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining mesh-free methods with the finite element method. </p> <div class="mw-heading mw-heading3"><h3 id="Spectral_element_method">Spectral element method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=28" title="Edit section: Spectral element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spectral_element_method" title="Spectral element method">Spectral element method</a></div><p>Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak-form partial equations based on high-order Lagrangian interpolants and used only with certain quadrature rules.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading3"><h3 id="Meshfree_methods">Meshfree methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=29" title="Edit section: Meshfree methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Meshfree_methods" title="Meshfree methods">Meshfree methods</a></div> <div class="mw-heading mw-heading3"><h3 id="Discontinuous_Galerkin_methods">Discontinuous Galerkin methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=30" title="Edit section: Discontinuous Galerkin methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin method</a></div> <div class="mw-heading mw-heading3"><h3 id="Finite_element_limit_analysis">Finite element limit analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=31" title="Edit section: Finite element limit analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_element_limit_analysis" title="Finite element limit analysis">Finite element limit analysis</a></div> <div class="mw-heading mw-heading3"><h3 id="Stretched_grid_method">Stretched grid method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=32" title="Edit section: Stretched grid method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stretched_grid_method" title="Stretched grid method">Stretched grid method</a></div> <div class="mw-heading mw-heading3"><h3 id="Loubignac_iteration">Loubignac iteration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=33" title="Edit section: Loubignac iteration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a> is an iterative method in finite element methods. </p> <div class="mw-heading mw-heading3"><h3 id="Crystal_plasticity_finite_element_method_(CPFEM)"><span id="Crystal_plasticity_finite_element_method_.28CPFEM.29"></span>Crystal plasticity finite element method (CPFEM)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=34" title="Edit section: Crystal plasticity finite element method (CPFEM)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates, which behave anisotropy under deformation, such as abnormal stress and strain localization. CPFEM, based on the slip (shear strain rate), can calculate dislocation, crystal orientation, and other texture information to consider crystal anisotropy during the routine. It has been applied in the numerical study of material deformation, surface roughness, fractures, etc. </p> <div class="mw-heading mw-heading3"><h3 id="Virtual_element_method_(VEM)"><span id="Virtual_element_method_.28VEM.29"></span>Virtual element method (VEM)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=35" title="Edit section: Virtual element method (VEM)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013)<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> as an extension of <a href="/wiki/Mimesis_(mathematics)" title="Mimesis (mathematics)">mimetic</a> <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference</a> (MFD) methods, is a generalization of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> in 3D) that are highly irregular and non-convex in shape. The name <i>virtual</i> derives from the fact that knowledge of the local shape function basis is not required and is, in fact, never explicitly calculated. </p> <div class="mw-heading mw-heading2"><h2 id="Link_with_the_gradient_discretization_method">Link with the gradient discretization method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=36" title="Edit section: Link with the gradient discretization method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the <a href="/wiki/Gradient_discretization_method" class="mw-redirect" title="Gradient discretization method">gradient discretization method</a> (GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and nonlinear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular FEMs. </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_to_the_finite_difference_method">Comparison to the finite difference method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=37" title="Edit section: Comparison to the finite difference method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference method</a> (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are: </p> <ul><li>The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_22-0" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>FDM is not usually used for irregular CAD geometries but more often for rectangular or block-shaped models.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>FEM generally allows for more flexible mesh adaptivity than FDM.<sup id="cite_ref-:1_22-1" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>The most attractive feature of finite differences is that it is straightforward to implement.<sup id="cite_ref-:1_22-2" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>One could consider the FDM a particular case of the FEM approach in several ways. E.g., first-order FEM is identical to FDM for <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a> if the problem is <a href="/wiki/Discretization" title="Discretization">discretized</a> by a regular rectangular mesh with each rectangle divided into two triangles.</li> <li>There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.</li> <li>The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is highly problem-dependent, and several examples to the contrary can be provided.</li></ul> <p>Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e., solving for deformation and stresses in solid bodies or dynamics of structures). In contrast, <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD) tend to use FDM or other methods like <a href="/wiki/Finite_volume_method" title="Finite volume method">finite volume method</a> (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more). Therefore the cost of the solution favors simpler, lower-order approximation within each cell. This is especially true for 'external flow' problems, like airflow around the car, airplane, or weather simulation. </p> <div class="mw-heading mw-heading2"><h2 id="Finite_element_and_fast_fourier_transform_(FFT)_methods"><span id="Finite_element_and_fast_fourier_transform_.28FFT.29_methods"></span>Finite element and fast fourier transform (FFT) methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=38" title="Edit section: Finite element and fast fourier transform (FFT) methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another method used for approximating solutions to a partial differential equation is the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier Transform</a> (FFT), where the solution is approximated by a fourier series computed using the FFT. For approximating the mechanical response of materials under stress, FFT is often much faster,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> but FEM may be more accurate.<sup id="cite_ref-:2_25-0" class="reference"><a href="#cite_note-:2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> One example of the respective advantages of the two methods is in simulation of <a href="/wiki/Rolling_(metalworking)" title="Rolling (metalworking)">rolling</a> a sheet of <a href="/wiki/Aluminium" title="Aluminium">aluminum</a> (an FCC metal), and <a href="/wiki/Wire_drawing" title="Wire drawing">drawing</a> a wire of <a href="/wiki/Tungsten" title="Tungsten">tungsten</a> (a BCC metal). This simulation did not have a sophisticated shape update algorithm for the FFT method. In both cases, the FFT method was more than 10 times as fast as FEM, but in the wire drawing simulation, where there were large deformations in <a href="/wiki/Crystallite" title="Crystallite">grains</a>, the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar.<sup id="cite_ref-:2_25-1" class="reference"><a href="#cite_note-:2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> FFT has a larger speed advantage in cases where the boundary conditions are given in the materials <a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">strain</a>, and loses some of its efficiency in cases where the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> is used to apply the boundary conditions, as more iterations of the method are needed.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The FE and FFT methods can also be combined in a <a href="/wiki/Voxel" title="Voxel">voxel</a> based method (2) to simulate deformation in materials, where the FE method is used for the macroscale stress and deformation, and the FFT method is used on the microscale to deal with the effects of microscale on the mechanical response.<sup id="cite_ref-:3_27-0" class="reference"><a href="#cite_note-:3-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Unlike FEM, FFT methods’ similarities to image processing methods means that an actual image of the microstructure from a microscope can be input to the solver to get a more accurate stress response. Using a real image with FFT avoids meshing the microstructure, which would be required if using FEM simulation of the microstructure, and might be difficult. Because fourier approximations are inherently periodic, FFT can only be used in cases of periodic microstructure, but this is common in real materials.<sup id="cite_ref-:3_27-1" class="reference"><a href="#cite_note-:3-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> FFT can also be combined with FEM methods by using fourier components as the variational basis for approximating the fields inside an element, which can take advantage of the speed of FFT based solvers.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Application">Application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=39" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vanadis_a1_test.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Vanadis_a1_test.gif/245px-Vanadis_a1_test.gif" decoding="async" width="245" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Vanadis_a1_test.gif/368px-Vanadis_a1_test.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Vanadis_a1_test.gif/490px-Vanadis_a1_test.gif 2x" data-file-width="511" data-file-height="157" /></a><figcaption>3D pollution transport model - concentration field on ground level</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vanadis_a2_test.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Vanadis_a2_test.gif/245px-Vanadis_a2_test.gif" decoding="async" width="245" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Vanadis_a2_test.gif/368px-Vanadis_a2_test.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c5/Vanadis_a2_test.gif 2x" data-file-width="466" data-file-height="151" /></a><figcaption>3D pollution transport model - concentration field on perpendicular surface </figcaption></figure> <p>Various specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and minimizing weight, materials, and costs.<sup id="cite_ref-Engineering_Asset_Management_29-0" class="reference"><a href="#cite_note-Engineering_Asset_Management-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>FEM allows detailed visualization of where structures bend or twist, indicating the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of modeling and system analysis. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured. The mesh is an integral part of the model and must be controlled carefully to give the best results. Generally, the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge, and further mesh refinement does not increase accuracy.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Human_knee_joint_FE_model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Human_knee_joint_FE_model.png/245px-Human_knee_joint_FE_model.png" decoding="async" width="245" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Human_knee_joint_FE_model.png/368px-Human_knee_joint_FE_model.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Human_knee_joint_FE_model.png/490px-Human_knee_joint_FE_model.png 2x" data-file-width="1266" data-file-height="640" /></a><figcaption>Finite Element Model of a human knee joint<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology in many industrial applications.<sup id="cite_ref-Hastings_32-0" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The introduction of FEM has substantially decreased the time to take products from concept to the production line.<sup id="cite_ref-Hastings_32-1" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Testing and development have been accelerated primarily through improved initial prototype designs using FEM.<sup id="cite_ref-McLaren-Mercedes_33-0" class="reference"><a href="#cite_note-McLaren-Mercedes-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.<sup id="cite_ref-Hastings_32-2" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>In the 1990s FEM was proposed for use in stochastic modeling for numerically solving probability models<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> and later for reliability assessment.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>FEM is widely applied for approximating differential equations that describe physical systems. This method is very popular in the community of <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">Computational fluid dynamics</a>, and there are many applications for solving <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> with FEM.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Recently, the application of FEM has been increasing in the researches of computational plasma. Promising numerical results using FEM for <a href="/wiki/Magnetohydrodynamics" title="Magnetohydrodynamics">Magnetohydrodynamics</a>, <a href="/wiki/Vlasov_equation" title="Vlasov equation">Vlasov equation</a>, and <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> have been proposed.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=40" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Applied_element_method" title="Applied element method">Applied element method</a></li> <li><a href="/wiki/Boundary_element_method" title="Boundary element method">Boundary element method</a></li> <li><a href="/wiki/C%C3%A9a%27s_lemma" title="Céa's lemma">Céa's lemma</a></li> <li><a href="/wiki/Computer_experiment" title="Computer experiment">Computer experiment</a></li> <li><a href="/wiki/Direct_stiffness_method" title="Direct stiffness method">Direct stiffness method</a></li> <li><a href="/wiki/Discontinuity_layout_optimization" title="Discontinuity layout optimization">Discontinuity layout optimization</a></li> <li><a href="/wiki/Discrete_element_method" title="Discrete element method">Discrete element method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Finite_element_machine" title="Finite element machine">Finite element machine</a></li> <li><a href="/wiki/Finite_element_method_in_structural_mechanics" title="Finite element method in structural mechanics">Finite element method in structural mechanics</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Finite_volume_method_for_unsteady_flow" title="Finite volume method for unsteady flow">Finite volume method for unsteady flow</a></li> <li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element method</a></li> <li><a href="/wiki/Interval_finite_element" title="Interval finite element">Interval finite element</a></li> <li><a href="/wiki/Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li> <li><a href="/wiki/Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann methods</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">List of finite element software packages</a></li> <li><a href="/wiki/Meshfree_methods" title="Meshfree methods">Meshfree methods</a></li> <li><a href="/wiki/Movable_cellular_automaton" title="Movable cellular automaton">Movable cellular automaton</a></li> <li><a href="/wiki/Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">Multidisciplinary design optimization</a></li> <li><a href="/wiki/Multiphysics" class="mw-redirect" title="Multiphysics">Multiphysics</a></li> <li><a href="/wiki/Patch_test_(finite_elements)" title="Patch test (finite elements)">Patch test</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Ritz_method" title="Rayleigh–Ritz method">Rayleigh–Ritz method</a></li> <li><a href="/wiki/Space_mapping" title="Space mapping">Space mapping</a></li> <li><a href="/wiki/STRAND7" title="STRAND7">STRAND7</a></li> <li><a href="/wiki/Tessellation_(computer_graphics)" title="Tessellation (computer graphics)">Tessellation (computer graphics)</a></li> <li><a href="/wiki/Weakened_weak_form" title="Weakened weak form">Weakened weak form</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method&action=edit&section=41" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHoangSchwab2005" class="citation journal cs1">Hoang, Viet Ha; Schwab, Christoph (2005). "High-dimensional finite elements for elliptic problems with multiple scales". <i>Multiscale Modeling & Simulation</i>. <b>3</b> (1). SIAM: 168–194. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F040606041">10.1137/040606041</a> (inactive 2024-11-02).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Multiscale+Modeling+%26+Simulation&rft.atitle=High-dimensional+finite+elements+for+elliptic+problems+with+multiple+scales&rft.volume=3&rft.issue=1&rft.pages=168-194&rft.date=2005&rft_id=info%3Adoi%2F10.1137%2F040606041&rft.aulast=Hoang&rft.aufirst=Viet+Ha&rft.au=Schwab%2C+Christoph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+element+method" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaryl_L._Logan2011" class="citation book cs1">Daryl L. Logan (2011). <i>A first course in the finite element method</i>. Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780495668275" title="Special:BookSources/9780495668275"><bdi>9780495668275</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+the+finite+element+method&rft.pub=Cengage+Learning&rft.date=2011&rft.isbn=9780495668275&rft.au=Daryl+L.+Logan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+element+method" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReddy2006" class="citation book cs1">Reddy, J. N. (2006). <i>An Introduction to the Finite Element Method</i> (Third ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780071267618" title="Special:BookSources/9780071267618"><bdi>9780071267618</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Finite+Element+Method&rft.edition=Third&rft.pub=McGraw-Hill&rft.date=2006&rft.isbn=9780071267618&rft.aulast=Reddy&rft.aufirst=J.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+element+method" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuebner2001" class="citation book cs1">Huebner, Kenneth H. (2001). <i>The Finite Element Method for Engineers</i>. 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Allaire and A. Craig: <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=HIwSDAAAQBAJ&q=%22finite+element%22">Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation</a></i>.</li> <li>K. J. Bathe: <i>Numerical methods in finite element analysis</i>, Prentice-Hall (1976).</li> <li>Thomas J.R. Hughes: <i>The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,</i> Prentice-Hall (1987).</li> <li>J. Chaskalovic: <i>Finite Elements Methods for Engineering Sciences</i>, Springer Verlag, (2008).</li> <li><a href="/wiki/Endre_S%C3%BCli" title="Endre Süli">Endre Süli</a>: <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/suli/fem.pdf"><i>Finite Element Methods for Partial Differential Equations</i></a>.</li> <li>O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu : <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=YocoaH8lnx8C">The Finite Element Method: Its Basis and Fundamentals</a></i>, Butterworth-Heinemann (2005).</li> <li>N. Ottosen, H. Petersson: <i>Introduction to the Finite Element Method, </i> Prentice-Hall (1992).</li> <li>Susanne C. Brenner, L. Ridgway Scott: <i>The Mathematical Theory of Finite Element Methods</i>, Springer-Verlag New York, ISBN 978-0-387-75933-3 (2008).</li> <li>Zohdi, T. I. (2018) A finite element primer for beginners-extended version including sample tests and projects. Second Edition <a rel="nofollow" class="external free" href="https://link.springer.com/book/10.1007/978-3-319-70428-9">https://link.springer.com/book/10.1007/978-3-319-70428-9</a></li> <li>Leszek F. Demkowicz: <i>Mathematical Theory of Finite Elements</i>, SIAM, ISBN 978-1-61197-772-1 (2024).</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output 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style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">Parabolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/FTCS_scheme" title="FTCS scheme">Forward-time central-space</a> (FTCS)</li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lax%E2%80%93Friedrichs_method" title="Lax–Friedrichs method">Lax–Friedrichs</a></li> <li><a href="/wiki/Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff</a></li> <li><a href="/wiki/MacCormack_method" title="MacCormack method">MacCormack</a></li> <li><a href="/wiki/Upwind_scheme" title="Upwind scheme">Upwind</a></li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_direction_implicit_method" class="mw-redirect" title="Alternating direction implicit method">Alternating direction-implicit</a> (ADI)</li> <li><a href="/wiki/Finite-difference_frequency-domain_method" title="Finite-difference frequency-domain method">Finite-difference frequency-domain</a> (FDFD)</li> <li><a href="/wiki/Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain</a> (FDTD)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Godunov%27s_scheme" title="Godunov's scheme">Godunov</a></li> <li><a href="/wiki/High-resolution_scheme" title="High-resolution scheme">High-resolution</a></li> <li><a href="/wiki/MUSCL_scheme" title="MUSCL scheme">Monotonic upstream-centered</a> (MUSCL)</li> <li><a href="/wiki/AUSM" class="mw-redirect" title="AUSM">Advection upstream-splitting</a> (AUSM)</li> <li><a href="/wiki/Riemann_solver" title="Riemann solver">Riemann solver</a></li> <li><a href="/wiki/ENO_methods" title="ENO methods">Essentially non-oscillatory</a> (ENO)</li> <li><a href="/wiki/WENO_methods" title="WENO methods">Weighted essentially non-oscillatory</a> (WENO)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Finite element</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a></li> <li><a href="/wiki/Extended_finite_element_method" title="Extended finite element method">Extended</a> (XFEM)</li> <li><a href="/wiki/Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin</a> (DG)</li> <li><a href="/wiki/Spectral_element_method" title="Spectral element method">Spectral element</a> (SEM)</li> <li><a href="/wiki/Mortar_methods" title="Mortar methods">Mortar</a></li> <li><a href="/wiki/Gradient_discretisation_method" title="Gradient discretisation method">Gradient discretisation</a> (GDM)</li> <li><a href="/wiki/Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a></li> <li><a href="/wiki/Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed</a> (S-FEM)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Meshfree_methods" title="Meshfree methods">Meshless/Meshfree</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a> (SPH)</li> <li><a href="/wiki/Peridynamics" title="Peridynamics">Peridynamics</a> (PD)</li> <li><a href="/wiki/Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a> (MPS)</li> <li><a href="/wiki/Material_point_method" title="Material point method">Material point method</a> (MPM)</li> <li><a href="/wiki/Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a> (PIC)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Domain_decomposition_methods" title="Domain decomposition methods">Domain decomposition</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schur_complement_method" title="Schur complement method">Schur complement</a></li> <li><a href="/wiki/Fictitious_domain_method" title="Fictitious domain method">Fictitious domain</a></li> <li><a href="/wiki/Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating</a> <ul><li><a href="/wiki/Additive_Schwarz_method" title="Additive Schwarz method">additive</a></li> <li><a href="/wiki/Abstract_additive_Schwarz_method" title="Abstract additive Schwarz method">abstract additive</a></li></ul></li> <li><a href="/wiki/Neumann%E2%80%93Dirichlet_method" title="Neumann–Dirichlet method">Neumann–Dirichlet</a></li> <li><a href="/wiki/Neumann%E2%80%93Neumann_methods" title="Neumann–Neumann methods">Neumann–Neumann</a></li> <li><a href="/wiki/Poincar%C3%A9%E2%80%93Steklov_operator" title="Poincaré–Steklov operator">Poincaré–Steklov operator</a></li> <li><a href="/wiki/Balancing_domain_decomposition_method" title="Balancing domain decomposition method">Balancing</a> (BDD)</li> <li><a href="/wiki/BDDC" title="BDDC">Balancing by constraints</a> (BDDC)</li> <li><a href="/wiki/FETI" title="FETI">Tearing and interconnect</a> (FETI)</li> <li><a href="/wiki/FETI-DP" title="FETI-DP">FETI-DP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_method" title="Spectral method">Spectral</a></li> <li><a href="/wiki/Pseudo-spectral_method" title="Pseudo-spectral method">Pseudospectral</a> (DVR)</li> <li><a href="/wiki/Method_of_lines" title="Method of lines">Method of lines</a></li> <li><a href="/wiki/Multigrid_method" title="Multigrid method">Multigrid</a></li> <li><a href="/wiki/Collocation_method" title="Collocation method">Collocation</a></li> <li><a href="/wiki/Level-set_method" title="Level-set method">Level-set</a></li> <li><a href="/wiki/Boundary_element_method" title="Boundary element method">Boundary element</a> <ul><li><a href="/wiki/Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments</a></li></ul></li> <li><a href="/wiki/Immersed_boundary_method" title="Immersed boundary method">Immersed boundary</a></li> <li><a href="/wiki/Analytic_element_method" title="Analytic element method">Analytic element</a></li> <li><a href="/wiki/Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li> <li><a href="/wiki/Infinite_difference_method" title="Infinite difference method">Infinite difference method</a></li> <li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin method</a></li></ul></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li> <li><a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li> <li><a href="/wiki/Integrable_algorithm" title="Integrable algorithm">Integrable algorithm</a></li> <li><a href="/wiki/Method_of_fundamental_solutions" title="Method of fundamental solutions">Method of fundamental solutions</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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