Document Zbl 0974.76043

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K.</a></div> <h2 class="title"> <strong>A posteriori error estimation for finite element solutions of Helmholtz’ equation. II: Estimation of the pollution error.</strong> <i>(English)</i> <a class="label nowrap" href="/0974.76043">Zbl 0974.76043</a> </h2> <div class="source"> <a href="/serials/462" title="Journal Profile">Int. J. Numer. Methods Eng.</a> <a href="/?q=in%3A43988" title="Articles in this Issue">40, No. 21, 3883-3900 (1997)</a>. </div> <div class="abstract">Summary: In part I of this investigation [see the foregoing entry], we proved that the standard a posteriori estimates, based only on local computations, may severely underestimate the exact error for the classes of wave-numbers and the types of meshes employed in engineering analyses. We showed that this is due to the fact that the local estimators do not measure the pollution effect inherent to the finite elements solutions of Helmholtz equation with large wavenumber. Here, we construct a posteriori estimates of the pollution error. We demonstrate that these estimates are reliable and can be used to correct the standard a posteriori error estimates in any patch of elements of interest.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1143744">Cited in <strong>31</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A76Q05" title="MSC2020">76Q05</a> </td> <td class="space"> Hydro- and aero-acoustics </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%3A65N30" title="MSC2020">65N30</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aa+posteriori+error+estimation">a posteriori error estimation</a>; <a href="/?q=ut%3Afinite+element+method">finite element method</a>; <a href="/?q=ut%3Aacoustic+scattering">acoustic scattering</a>; <a href="/?q=ut%3AHelmholtz+equation">Helmholtz equation</a>; <a href="/?q=ut%3Alarge+wavenumber">large wavenumber</a>; <a href="/?q=ut%3Apollution+error">pollution error</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 0974.76043" data-ciurl="/ci/01143744" data-biburl="/bibtex/01143744.bib" data-amsurl="/amsrefs/01143744.bib" data-xmlurl="/xml/01143744.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01143744.pdf" title="Zbl 0974.76043 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/10.1002/(SICI)1097-0207(19971115)40:21<3883::AID-NME231>3.0.CO;2-V" aria-label="DOI for “A posteriori error estimation for finite element solutions of Helmholtz' equation. II: Estimation of the pollution error”" title="10.1002/(SICI)1097-0207(19971115)40:21<3883::AID-NME231>3.0.CO;2-V">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">Bayliss, J. Comput. Phys. 59 pp 396– (1985) · <a href="/0647.65072" class="nowrap">Zbl 0647.65072</a> · <a href="https://doi.org/10.1016/0021-9991(85)90119-6" class="nowrap">doi:10.1016/0021-9991(85)90119-6</a></td> </tr><tr> <td>[2]</td> <td class="space">and , ’Accuracy control for acoustic finite element analysis’, ACUSTICA I, 1996, p. S156.</td> </tr><tr> <td>[3]</td> <td class="space">Ihlenburg, Comput. Math. Appl. 30 pp 9– (1995) · <a href="/0838.65108" class="nowrap">Zbl 0838.65108</a> · <a href="https://doi.org/10.1016/0898-1221(95)00144-N" class="nowrap">doi:10.1016/0898-1221(95)00144-N</a></td> </tr><tr> <td>[4]</td> <td class="space">Ihlenburg, Int. J. Numer. Meth. Engng. 38 pp 3745– (1995) · <a href="/0851.73062" class="nowrap">Zbl 0851.73062</a> · <a href="https://doi.org/10.1002/nme.1620382203" class="nowrap">doi:10.1002/nme.1620382203</a></td> </tr><tr> <td>[5]</td> <td class="space">and , ’Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wavenumbers’, Technical Note BN-1172, Institute for Physical Science and Technology, University of Maryland at College Park, 1994.</td> </tr><tr> <td>[6]</td> <td class="space">Babuška, Comput. Meth. Appl. Mech. Engng.</td> </tr><tr> <td>[7]</td> <td class="space">and , ’The pollution effect in FE solutions of the 3D-Helmholtz equation with large wave number’, TICAM report 6/96.</td> </tr><tr> <td>[8]</td> <td class="space">Ihlenburg, SIAM J. Numer. Anal. (1994)</td> </tr><tr> <td>[9]</td> <td class="space">and , ’The partition of unity method: basic theory and applications’, TICAM report 96-03, University of Texas, at Austin, 1996.</td> </tr><tr> <td>[10]</td> <td class="space">’On generalised finite element methods’, PhD dissertation, University of Maryland, 1995.</td> </tr><tr> <td>[11]</td> <td class="space">Ihlenburg, Computer Assisted Mechanics and Engineering Sciences</td> </tr><tr> <td>[12]</td> <td class="space">’Large scale electromagnetic analysis using unstructured grids’, Seminar lecture, TICAM, University of Texas, at Texas, 1996.</td> </tr><tr> <td>[13]</td> <td class="space">Babuška, Finite Elem. Anal. and Des. 17 pp 273– (1994) · <a href="https://doi.org/10.1016/0168-874X(94)90003-5" class="nowrap">doi:10.1016/0168-874X(94)90003-5</a></td> </tr><tr> <td>[14]</td> <td class="space">Babuška, Int. J. Numer. Meth. Engng. 38 pp 4207– (1995) · <a href="/0844.65078" class="nowrap">Zbl 0844.65078</a> · <a href="https://doi.org/10.1002/nme.1620382408" class="nowrap">doi:10.1002/nme.1620382408</a></td> </tr><tr> <td>[15]</td> <td class="space">Babuška, Comput. Methods Appl. Mech. Engng. 140 pp 1– (1997) · <a href="/0896.73055" class="nowrap">Zbl 0896.73055</a> · <a href="https://doi.org/10.1016/S0045-7825(96)01013-4" class="nowrap">doi:10.1016/S0045-7825(96)01013-4</a></td> </tr><tr> <td>[16]</td> <td class="space">Babuška, Comput. Methods Appl. Mech. Engng. 114 pp 307– (1994) · <a href="https://doi.org/10.1016/0045-7825(94)90177-5" class="nowrap">doi:10.1016/0045-7825(94)90177-5</a></td> </tr><tr> <td>[17]</td> <td class="space">Babuška, Int. J. Numer. Meth. Engng. 37 pp 1073– (1994) · <a href="/0811.65088" class="nowrap">Zbl 0811.65088</a> · <a href="https://doi.org/10.1002/nme.1620370702" class="nowrap">doi:10.1002/nme.1620370702</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2024 <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 = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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