Document Zbl 1017.65088

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Comput.</a> <a href="/?q=in%3A92922" title="Articles in this Issue">72, No. 242, 541-576 (2003)</a>. </div> <div class="abstract">This paper is motivated by the numerical simulation of flow transport in highly heterogeneous porous media. By using the most elaborated tools of mathematical finite element methods, the authors analyze a mixed multiscale finite element method with an over-sampling technique which is very well suited for solving second-order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems.<br class="zbmathjax-paragraph">By assuming that the oscillating coefficients are locally periodic and by using homogenization techniques, the authors prove the convergence of the method. Finally, they report some numerical experiments which demonstrate the efficiency and accuracy of the proposed method.<br class="zbmathjax-paragraph">This paper is nicely written and looks like a strong mathematical and. numerical contribution to the subject.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A5453">Michel Bernadou (Paris La Defense)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1871605">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A1871605">Cited in <strong>187</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%3A35B27" title="MSC2020">35B27</a> </td> <td class="space"> Homogenization in context of PDEs; PDEs in media with periodic structure </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%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%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%3A76M50" title="MSC2020">76M50</a> </td> <td class="space"> Homogenization applied to problems in fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76S05" title="MSC2020">76S05</a> </td> <td class="space"> Flows in porous media; filtration; seepage </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aflow+transport">flow transport</a>; <a href="/?q=ut%3Aporous+media">porous media</a>; <a href="/?q=ut%3Amixed+multiscale+finite+element+method">mixed multiscale finite element method</a>; <a href="/?q=ut%3Asecond-order+elliptic+equations">second-order elliptic equations</a>; <a href="/?q=ut%3Aoscillating+coefficients">oscillating coefficients</a>; <a href="/?q=ut%3Ahomogenization">homogenization</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Anumerical+experiments">numerical experiments</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 1017.65088" data-ciurl="/ci/01871605" data-biburl="/bibtex/01871605.bib" data-amsurl="/amsrefs/01871605.bib" data-xmlurl="/xml/01871605.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01871605.pdf" title="Zbl 1017.65088 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/10.1090/S0025-5718-02-01441-2" aria-label="DOI for “A mixed multiscale finite element method for elliptic problems with oscillating coefficients”" title="10.1090/S0025-5718-02-01441-2">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">T. 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