Document Zbl 1277.65092

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Berlin: Springer (ISBN 978-3-642-36518-8/hbk; 978-3-642-36519-5/ebook). xiv, 685 p. (2013). </div> <div class="abstract">The rough contents of this monograph are as follows: Chapter 1: Variational formulations and finite element methods, Chapter 2: Function spaces and finite element approximations, Chapter 3: Algebraic aspects of saddle point problems, Chapter 4: Saddle point problems in Hilbert spaces, Chapter 5: Approximation of saddle point problems, Chapter 6: Complements: Stabilization methods, eigenvalue problems, Chapter 7: Mixed methods for elliptic problems, Chapter 8: Incompressible materials and flow problems, Chapter 9: Complements on elasticity problems, Chapter 10: Complements on plate problems and Chapter 11: Mixed finite elements for electromagnetic problems. The book also contains a short Preface, References and an Index.<br class="zbmathjax-paragraph">The authors are leading experts in the mathematical foundation of the finite element method with outstanding publications in mixed and hybrid formulations of this method. The book is mainly built up on their results. Thus, the authors first review the basic results from functional analysis and linear algebra on which they construct the mixed finite element methods. The second-and fourth-order elliptic problems, which form the core of the work, are separately analyzed for the one-, two- and three-dimensional cases. The mixed (hybrid) finite element methods are compared with finite volume methods and various error estimates are provided. A particular attention is paid to eigenvalue problems. Mixed variational formulations are also provided for a large variety of problems from linear elasticity, incompressible Stokes flow and electromagnetism. Along with these formulations the authors introduce a fairly general and rigorous framework for the analysis of finite element approximations to problems involving incompressible type conditions. All in all, the book contains the fundamental results as well as a lot of usual and peculiar ingredients one needs to understand and efficiently practice the mixed and related finite element methods.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A5388">Calin Ioan Gheorghiu (Cluj-Napoca)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6154429">Cited in <strong>1197</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N30" title="MSC2020">65N30</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74S05" title="MSC2020">74S05</a> </td> <td class="space"> Finite element methods applied to problems in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M10" title="MSC2020">76M10</a> </td> <td class="space"> Finite element methods applied to problems in fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78M10" title="MSC2020">78M10</a> </td> <td class="space"> Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65-02" title="MSC2020">65-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to numerical analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N25" title="MSC2020">65N25</a> </td> <td class="space"> Numerical methods for eigenvalue problems for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N15" title="MSC2020">65N15</a> </td> <td class="space"> Error bounds for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N12" title="MSC2020">65N12</a> </td> <td class="space"> Stability and convergence of numerical methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J25" title="MSC2020">35J25</a> </td> <td class="space"> Boundary value problems for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74B05" title="MSC2020">74B05</a> </td> <td class="space"> Classical linear elasticity </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A25" title="MSC2020">78A25</a> </td> <td class="space"> Electromagnetic theory (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76D07" title="MSC2020">76D07</a> </td> <td class="space"> Stokes and related (Oseen, etc.) flows </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35P15" title="MSC2020">35P15</a> </td> <td class="space"> Estimates of eigenvalues in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J40" title="MSC2020">35J40</a> </td> <td class="space"> Boundary value problems for higher-order elliptic equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afinite+element+methods">finite element methods</a>; <a href="/?q=ut%3Asaddle+point+problems">saddle point problems</a>; <a href="/?q=ut%3Aeigenvalue+problems">eigenvalue problems</a>; <a href="/?q=ut%3Amixed+methods">mixed methods</a>; <a href="/?q=ut%3Alinear+elliptic+problems">linear elliptic problems</a>; <a href="/?q=ut%3Aincompressible+materials">incompressible materials</a>; <a href="/?q=ut%3AStokes+problem">Stokes problem</a>; <a href="/?q=ut%3Alinear+elasticity+problems">linear elasticity problems</a>; <a href="/?q=ut%3Aelectromagnetism+problems">electromagnetism problems</a>; <a href="/?q=ut%3Amonograph">monograph</a>; <a href="/?q=ut%3Afinite+volume+method">finite volume method</a>; <a href="/?q=ut%3Aerror+estimates">error estimates</a>; <a href="/?q=ut%3Aeigenvalue+problem">eigenvalue problem</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1277.65092" data-ciurl="/ci/06154429" data-biburl="/bibtex/06154429.bib" data-amsurl="/amsrefs/06154429.bib" data-xmlurl="/xml/06154429.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06154429.pdf" title="Zbl 1277.65092 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/978-3-642-36519-5" aria-label="DOI for “Mixed finite element methods and applications”" title="10.1007/978-3-642-36519-5">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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