Divergence free virtual elements for the stokes problem on polygonal meshes | ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)

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</script> <link title="schema(PRISM)" rel="schema.prism" href="http://prismstandard.org/namespaces/1.2/basic/"> <meta name="citation_journal_title" content="ESAIM: Mathematical Modelling and Numerical Analysis"> <meta name="citation_journal_abbrev" content="ESAIM: M2AN"> <meta name="citation_publisher" content="EDP Sciences"> <meta name="citation_author" content="Lourenco Beirão da Veiga"> <meta name="citation_author_institution" content="Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy."> <meta name="citation_author" content="Carlo Lovadina"> <meta name="citation_author_institution" content="Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy."> <meta name="citation_author_institution" content="IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy."> <meta name="citation_author" content="Giuseppe Vacca"> <meta name="citation_author_institution" content="Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy."> <meta name="citation_title" content="Divergence free virtual elements for the stokes problem on polygonal meshes"> <meta name="citation_online_date" content="2017/02/20"> <meta name="citation_publication_date" content="2017/03/01"> <meta name="citation_volume" content="51"> <meta name="citation_issue" content="2"> <meta name="citation_firstpage" content="509"> <meta name="citation_lastpage" content="535"> <meta name="citation_doi" content="10.1051/m2an/2016032"> <meta name="citation_abstract_html_url" content="https://www.esaim-m2an.org/articles/m2an/abs/2017/02/m2an150187/m2an150187.html"> <meta name="citation_pdf_url" content="https://www.esaim-m2an.org/articles/m2an/pdf/2017/02/m2an150187.pdf"> <meta name="citation_issn" content="0764-583X"> <meta name="citation_issn" content="1290-3841"> <meta name="citation_language" content="en"> <meta name="citation_keyword" content="Virtual element method"> <meta name="citation_keyword" content="polygonal meshes"> <meta name="citation_keyword" content="Stokes problem"> <meta name="citation_keyword" content="divergence free approximation"> <meta name="citation_article_type" content="Research Article"> <meta name="citation_abstract" lang="en" content="In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is <i>pointwise <i/>divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with <i>fewer <i/>degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice."> <meta name="citation_reference" content="citation_author=B. Ahmad;citation_author=A. Alsaedi;citation_author=F. Brezzi;citation_author=L.D. Marini;citation_author=A. Russo;citation_title=Equivalent projectors for virtual element methods;citation_publication_date=2013;citation_firstpage=376;citation_lastpage=391;"> <meta name="citation_reference" content="citation_author=P.F. Antonietti;citation_author=S. 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</strong>51, Number <strong>2</strong>, March-April 2017 </div> </td> <tr><td colspan="3" ></td></tr> <tr><td colspan="3" class="sep_issue"></td></tr> </tr> <tr> <th> Page(s) </th> <td></td> <td>509 - 535</td> </tr> <tr> <th>DOI</th> <td></td> <td> <a href="https://doi.org/10.1051/m2an/2016032"> https://doi.org/10.1051/m2an/2016032 </a> </td> </tr> <tr> <th>Published online</th> <td></td> <td> 20 February 2017 </td> </tr> </table> </div> <div id="article"> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"> <div id="contenu-min"> <div id="head"> ESAIM: M2AN 51 (2017) 509-535 <h2 class="title">Divergence free virtual elements for the stokes problem on polygonal meshes</h2> <div class="article-authors"><p class="bold"><span id="m2an150187-author-1" data-url="/component/author/?dkey=10.1051/m2an/2016032&n=1" class="author">Lourenco Beirão da Veiga</span><sup>1</sup>, <span id="m2an150187-author-2" data-url="/component/author/?dkey=10.1051/m2an/2016032&n=2" class="author">Carlo Lovadina</span><sup>2</sup><sup>,3</sup> and <span id="m2an150187-author-3" data-url="/component/author/?dkey=10.1051/m2an/2016032&n=3" class="author">Giuseppe Vacca</span><sup>4</sup></p></div> <p class="aff"><sup>1</sup> Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy. <br> <a href="mailto:lourenco.beirao@unimib.it">lourenco.beirao@unimib.it</a> <br><sup>2</sup> Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy. <br><sup>3</sup> IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy. <br> <a href="mailto:carlo.lovadina@unipv.it%20">carlo.lovadina@unipv.it </a> <br><sup>4</sup> Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy. <br> <a href="mailto:giuseppe.vacca@uniba.it">giuseppe.vacca@uniba.it</a> <br></p> <p class="history"><span class="bold">Received: </span>6 October 2015<br><span class="bold">Revised: </span>3 March 2016<br><span class="bold">Accepted: </span>21 April 2016</p> <p class="bold"><a name="abs"></a>Abstract</p> <p>In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is <i>pointwise </i>divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with <i>fewer </i>degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.</p> <div class="kword"> <p><span class="bold">Mathematics Subject Classification: </span>65N30 / 65N12 / 65N15 / 76D07</p> <p><span class="bold">Key words: </span>Virtual element method / polygonal meshes / Stokes problem / divergence free approximation</p> </div> </div> <p><i>© EDP Sciences, SMAI 2017</i></p> </div> </div> <div id="metrics-tabs" data-doi="10.1051/m2an/2016032" data-edps_ref="m2an150187"> <nav class="toolbar"> <button class="toolbar-item" id="metrics-siq">Current usage metrics</button> <button class="toolbar-item" id="info">About article metrics</button> <button class="toolbar-item" id="return">Return to article</button> </nav> <div class="panel" data-for="metrics-siq"> </div> <div class="panel" data-for="metrics-alm"></div> <div class="panel" data-for="info"> <p>Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.</p> <p>Data correspond to usage on the plateform after 2015. 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Divergence free virtual elements for the stokes problem on polygonal meshes | ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)