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Methods Partial Differ. Equations</a> <a href="/?q=in%3A367490" title="Articles in this Issue">33, No. 4, 1183-1207 (2017)</a>. </div> <div class="abstract"><div class="pre">The authors present a space-time continuous Galerkin (STCG) method with mesh modification for viscoelastic wave equations. They first establish the STCG scheme with mesh modification and demonstrate the existence, uniqueness and stability of the approximate solutions. Error estimates with global and local spatial mesh sizes in \( L^{\infty}(L^{2})\) norm are obtained without any space time mesh restrictions and provide global convergence rates under reasonable assumptions on the spatial mesh. A numerical example is given to illustrate the theory discussed.</div><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10126">Seenith Sivasundaram (Daytona Beach)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6763955">Cited in <strong>7</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L05" title="MSC2020">35L05</a> </td> <td class="space"> Wave equation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M12" title="MSC2020">65M12</a> </td> <td class="space"> Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M15" title="MSC2020">65M15</a> </td> <td class="space"> Error bounds for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anumerical+example">numerical example</a>; <a href="/?q=ut%3Aoptimal+order+error+estimates">optimal order error estimates</a>; <a href="/?q=ut%3Aspace-time+continuous+Galerkin+method">space-time continuous Galerkin method</a>; <a href="/?q=ut%3Aviscoelastic+wave+equations">viscoelastic wave equations</a>; <a href="/?q=ut%3Amesh+modification">mesh modification</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Aconvergence">convergence</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1373.65074" data-ciurl="/ci/06763955" data-biburl="/bibtex/06763955.bib" data-amsurl="/amsrefs/06763955.bib" data-xmlurl="/xml/06763955.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06763955.pdf" title="Zbl 1373.65074 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/num.22138" aria-label="DOI for “A space-time continuous Galerkin method with mesh modification for viscoelastic wave equations”" title="10.1002/num.22138">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">Gurtin, M.; Pipkin, A., A general theory of heat conduction with finite wave speeds, Arch Ration Mech Anal, 31, 113-126, 1968 · <a href="/0164.12901" class="nowrap">Zbl 0164.12901</a></td> </tr><tr> <td>[2]</td> <td class="space">Yuan, Y. 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E., Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer Math, 34, 41-62, 1980 · <a href="/0404.65055" class="nowrap">Zbl 0404.65055</a></td> </tr><tr> <td>[19]</td> <td class="space">Bangerth, W.; Rannacher, R., Finite element approximation of the acoustic wave equation: error control and mesh adaptation, East‐West J Numer Math, 7, 263-282, 1999 · <a href="/0948.65098" class="nowrap">Zbl 0948.65098</a></td> </tr><tr> <td>[20]</td> <td class="space">Jimack, P. K.; M. J. Baines and (ed.); K. W. Morton (ed.), A new approach to finite element error control for time‐dependent, Numerical methods for fluid dynamics, Vol. 4, 567-573, 1993, Oxford University Press: Oxford · <a href="/0801.76045" class="nowrap">Zbl 0801.76045</a></td> </tr><tr> <td>[21]</td> <td class="space">Lohner, R., An adaptive finite element scheme for transient problems in CFD, Comput Methods Appl Mech Eng, 50, 323-338, 1987 · <a href="/0611.73079" class="nowrap">Zbl 0611.73079</a></td> </tr><tr> <td>[22]</td> <td class="space">Aziz, A. K.; Monk, P., Continuous finite elements in space and time for the Heat equation, Math Comp, 52, 255-274, 1989 · <a href="/0673.65070" class="nowrap">Zbl 0673.65070</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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