Document Zbl 1158.65078

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Comput. Methods Eng.</a> <a href="/?q=in%3A216251" title="Articles in this Issue">15, No. 1, 37-66 (2008)</a>. </div> <div class="abstract">The Helmholtz equation $\nabla ^{2}u(\mathbf{x})+\kappa ^{2}u(\mathbf{x})=h( \mathbf{x}),$ where $\nabla ^{2}$ is the Laplacian, $\kappa $ is the wave number, $h$ is a forcing function and $u$ is the amplitude, finds applications in many important fields including aeroacoustics, under-water acoustics, seismic inversion and electromagnetics. Therefore computation of its solutions in a two or three dimensional domain is important. The linear system arising from a discretization of the Helmholtz equation is typically characterized by indefiniteness of the (real part of the) eigenvalues of the corresponding coefficient matrix, and hence the corresponding iterative scheme may encounter convergence problems. <br class="zbmathjax-paragraph">This paper reviews and highlights some recent advances in iterative methods for the Helmholtz equation. In particular, the author focuses on the Krylov subspace methods and the shifted Laplacian preconditioner. Some theories behind the shifted Laplacian preconditioner are given, and numerical results are presented for realistic problems. There are 142 references listed in this paper which include several recent surveys on similar subjects. The emphasis is on engineering computation, however, and the reader who is interested in the theoretical aspects is therefore encouraged to look for additional sources as well.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A9674">Sui Sun Cheng (Hsinchu)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5298423">Cited in <strong>71</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N06" title="MSC2020">65N06</a> </td> <td class="space"> Finite difference methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F10" title="MSC2020">65F10</a> </td> <td class="space"> Iterative numerical methods for linear systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N55" title="MSC2020">65N55</a> </td> <td class="space"> Multigrid methods; domain decomposition for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F35" title="MSC2020">65F35</a> </td> <td class="space"> Numerical computation of matrix norms, conditioning, scaling </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J05" title="MSC2020">35J05</a> </td> <td class="space"> Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHelmholtz+equation">Helmholtz equation</a>; <a href="/?q=ut%3Aiterative+methods">iterative methods</a>; <a href="/?q=ut%3AKrylov+subspace+methods">Krylov subspace methods</a>; <a href="/?q=ut%3Apreconditioner">preconditioner</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</a>; <a href="/?q=ut%3Afinite+difference+method">finite difference method</a></div> <div class="software"> <h3>Software:</h3><a href="/software/679">PARDISO</a>; <a href="/software/19092">LFA</a>; <a href="/software/8225">CGS</a>; <a href="/software/9677">Wesseling</a>; <a href="/software/8566">symrcm</a>; <a href="/software/8734">ILUT</a>; <a href="/software/41207">AILU</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 1158.65078" data-ciurl="/ci/05298423" data-biburl="/bibtex/05298423.bib" data-amsurl="/amsrefs/05298423.bib" data-xmlurl="/xml/05298423.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05298423.pdf" title="Zbl 1158.65078 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/10.1007/s11831-007-9013-7" aria-label="DOI for “Advances in iterative methods and preconditioners for the Helmholtz equation”" title="10.1007/s11831-007-9013-7">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">Abarbanel S, Gottlieb D (1997) A mathematical analysis of the PML method. 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