Document Zbl 0822.65078

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W.</a>; <a href="/authors/stynes.martin" title="Author Profile">Stynes, M.</a></div> <h2 class="title"> <strong>An analysis of the cell vertex method.</strong> <i>(English)</i> <a class="label nowrap" href="/0822.65078">Zbl 0822.65078</a> </h2> <div class="source"> <a href="/serials/1047" title="Journal Profile">RAIRO, Modélisation Math. Anal. Numér.</a> <a href="/?q=in%3A25630" title="Articles in this Issue">28, No. 6, 699-724 (1994)</a>. </div> <div class="abstract">The authors propose an analysis of the cell vertex scheme, based on mesh- dependent norms in a nonconforming Petrov-Galerkin framework. A mapping is introduced between the trial and test spaces similar to the upwinded test functions of finite element methods or the upwinded control volumes of finite volume methods.<br class="zbmathjax-paragraph">Particular mappings are used to analyze the one-dimensional convection- diffusion problem. For pure convection in two dimensions a new natural seminorm is used to prove local and global error estimates for the cases of flow transverse and parallel to the grid.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4787">E.Krause (Aachen)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A754222">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%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%3A35J25" title="MSC2020">35J25</a> </td> <td class="space"> Boundary value problems for second-order elliptic equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acell+vertex+method">cell vertex method</a>; <a href="/?q=ut%3Anonconforming+Petrov-Galerkin+method">nonconforming Petrov-Galerkin method</a>; <a href="/?q=ut%3Afinite+element+methods">finite element methods</a>; <a href="/?q=ut%3Afinite+volume+methods">finite volume methods</a>; <a href="/?q=ut%3Aconvection-diffusion+problem">convection-diffusion problem</a>; <a href="/?q=ut%3Aerror+estimates">error estimates</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 0822.65078" data-ciurl="/ci/00754222" data-biburl="/bibtex/00754222.bib" data-amsurl="/amsrefs/00754222.bib" data-xmlurl="/xml/00754222.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00754222.pdf" title="Zbl 0822.65078 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/10.1051/m2an/1994280606991" aria-label="DOI for “An analysis of the cell vertex method”" title="10.1051/m2an/1994280606991">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eudml.org/doc/193756" title="Open Access at EuDML">EuDML</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">J. W. BARRETT and K. W. MORTON, 1984, Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Computer Methods in Applied Mechanics and Engineering, 45, 97-122. Zbl0562.76086 MR759805 · <a href="/0562.76086" class="nowrap">Zbl 0562.76086</a> · <a href="https://doi.org/10.1016/0045-7825(84)90152-X" class="nowrap">doi:10.1016/0045-7825(84)90152-X</a></td> </tr><tr> <td>[2]</td> <td class="space">P. W. HEMKER, 1977, A numerical study of stiff two-point boundary problems, PhD thesis, Mathematisch Centrum, Amsterdam. Zbl0426.65043 MR488784 · <a href="/0426.65043" class="nowrap">Zbl 0426.65043</a></td> </tr><tr> <td>[3]</td> <td class="space">R. B. KELLOGG and A. TSAN, 1978, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation, 32, 1025-1039. Zbl0418.65040 MR483484 · <a href="/0418.65040" class="nowrap">Zbl 0418.65040</a> · <a href="https://doi.org/10.2307/2006331" class="nowrap">doi:10.2307/2006331</a></td> </tr><tr> <td>[4]</td> <td class="space">J. A. MACKENZIE, 1991, Cell vertex finite volume methods for the solution of the compressible Navier-Stokes equations, PhD thesis, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD.</td> </tr><tr> <td>[5]</td> <td class="space">J. A. MACKENZIE and K. W. MORTON, 1992Finite volume solutions of convection-diffusion test problems, Mathematics of Computation, 60(201), 189-220. Zbl0797.76072 MR1153168 · <a href="/0797.76072" class="nowrap">Zbl 0797.76072</a> · <a href="https://doi.org/10.2307/2153161" class="nowrap">doi:10.2307/2153161</a></td> </tr><tr> <td>[6]</td> <td class="space">K. W. MORTON, 1991, Finite volume methods and their analysis, in J. R. Whiteman, editor, The Mathematics of Finite Elements and Applications VII MAFELAP 1990, Academic Press, 189-214. Zbl0843.65070 MR1132499 · <a href="/0843.65070" class="nowrap">Zbl 0843.65070</a></td> </tr><tr> <td>[7]</td> <td class="space">K. W. MORTON, 1992, Upwinded test functions for finite element and finite volume methods, in D. F. Griffiths and G. A. Watson, editors, Numerical analysis 1991 Proceedings of the 14th Dundee Conference, June 1991, number 260 in Pitman Research Notes in Mathematics Series, pp. 128-141, Longman Scientific and Technical. Zbl0801.65096 MR1177232 · <a href="/0801.65096" class="nowrap">Zbl 0801.65096</a></td> </tr><tr> <td>[8]</td> <td class="space">K. W. MORTON, P. I. CRUMPTON and J. A. MACKENZIE, 1993, Cell vertex methods for inviscid and viscous flows, Computers Fluids, 22(2/3), 91-102. Zbl0779.76073 · <a href="/0779.76073" class="nowrap">Zbl 0779.76073</a> · <a href="https://doi.org/10.1016/0045-7930(93)90042-8" class="nowrap">doi:10.1016/0045-7930(93)90042-8</a></td> </tr><tr> <td>[9]</td> <td class="space">K. W. MORTON and E. SÜLI, 1991, Finite volume methods and their analysis, IMA Journal of Numerical Analysis, 11, 241-260. Zbl0729.65087 MR1105229 · <a href="/0729.65087" class="nowrap">Zbl 0729.65087</a> · <a href="https://doi.org/10.1093/imanum/11.2.241" class="nowrap">doi:10.1093/imanum/11.2.241</a></td> </tr><tr> <td>[10]</td> <td class="space">M. STYNES and E. O’RIORDAN, 1991, An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques, Mathematics of Computation, 56, 663-676. Zbl0718.65062 MR1068809 · <a href="/0718.65062" class="nowrap">Zbl 0718.65062</a> · <a href="https://doi.org/10.2307/2008400" class="nowrap">doi:10.2307/2008400</a></td> </tr><tr> <td>[11]</td> <td class="space">E. SÜLI, 1991, The accuracy of finite volume methods on distorted partitions. In J. R. Whiteman, editor. The Proceedings of the Conference on The Mathematics of Finite Elements and Applications VII MAFELAP, pp. 253-260. Academic Press. MR1132503</td> </tr><tr> <td>[12]</td> <td class="space">E. SÜLI, 1992, The accuracy of cell vertex finite volume methods on quadrilateral meshes, Mathematics of Computation, 59(200), 359-382. Zbl0767.65072 MR1134740 · <a href="/0767.65072" class="nowrap">Zbl 0767.65072</a> · <a href="https://doi.org/10.2307/2153062" class="nowrap">doi:10.2307/2153062</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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