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Comput. Phys.</a> <a href="/?q=in%3A53726" title="Articles in this Issue">142, No. 1, 163-181 (1998)</a>. </div> <div class="abstract">The authors examine two types of preconditioners for the discrete indefinite Helmholtz equation with Sommerfeld-like boundary conditions. The first is derived by discretization of a related continuous operator, the second uses the block Toeplitz approximation to the desired problem. The resulting preconditioning matrices allow the use of fast transform methods (e.g. fast Fourier transform) and differ from the discrete Helmholtz operator by an operator of low rank. Some numerical experiments presented in the paper demonstrate the efficiency of the method when combined with Krylov subspace iteration. The authors show that the performance of restared GMRES with the proposed preconditioners is relatevely insensitive to the discretization mesh size and the wave number, and the algorithms are highly parallelizable. The presented technique is potentially applicable to inhomogeneous media, exterior domain problems and non-Cartesian grids.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7984">Mária Lukáčová (Brno)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1290137">Cited in <strong>18</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%3A35J05" title="MSC2020">35J05</a> </td> <td class="space"> Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation </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%3A65F35" title="MSC2020">65F35</a> </td> <td class="space"> Numerical computation of matrix norms, conditioning, scaling </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHelmholtz+equation">Helmholtz equation</a>; <a href="/?q=ut%3ASommerfeld-like+boundary+conditions">Sommerfeld-like boundary conditions</a>; <a href="/?q=ut%3Apreconditioners">preconditioners</a>; <a href="/?q=ut%3AKrylov+subspace+iterations">Krylov subspace iterations</a>; <a href="/?q=ut%3Aparallel+computation">parallel computation</a>; <a href="/?q=ut%3Anumerical+experiments">numerical experiments</a></div> <div class="software"> <h3>Software:</h3><a href="/software/6496">MPI</a>; <a href="/software/503">LAPACK</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 0929.65089" data-ciurl="/ci/01290137" data-biburl="/bibtex/01290137.bib" data-amsurl="/amsrefs/01290137.bib" data-xmlurl="/xml/01290137.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01290137.pdf" title="Zbl 0929.65089 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/10.1006/jcph.1998.5939" aria-label="DOI for “Efficient iterative solution of the three-dimensional Helmholtz equation”" title="10.1006/jcph.1998.5939">DOI</a> <a class="btn btn-default btn-xs" type="button" href="http://hdl.handle.net/1903/912" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anderson, E., LAPACK Users’ Guide (1995) · <a href="/0755.65028" class="nowrap">Zbl 0755.65028</a></td> </tr><tr> <td>[2]</td> <td class="space">Barrett, R., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (1993) · <a href="/0814.65030" class="nowrap">Zbl 0814.65030</a></td> </tr><tr> <td>[3]</td> <td class="space">Bayliss, A.; Goldstein, C. 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