Approximation theory methods for solving elliptic eigenvalue problems
Identifieur interne : 001342 ( Istex/Corpus ); précédent : 001341; suivant : 001343Approximation theory methods for solving elliptic eigenvalue problems
Auteurs : G. StillSource :
- ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik [ 0044-2267 ] ; 2003-07-07.
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Abstract
Eigenvalue problems with elliptic operators L on a domain G ⊂ R2 are considered. By applying results from complex approximation theory we obtain results on the approximation properties of special classes of solutions of Lu = 0 on G . These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a‐posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics (‘relaxed plasma’, ‘MHD‐equation’).
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DOI: 10.1002/zamm.200310081
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Box 217, 7500 AE Enschede, The Netherlands</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: g.j.still@math.utwente.nl</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:4EED5B54B8C087DCD6BCBA275CB1EBEFBBB6693D</idno><date when="2003" year="2003">2003</date><idno type="doi">10.1002/zamm.200310081</idno><idno type="url">https://api.istex.fr/document/4EED5B54B8C087DCD6BCBA275CB1EBEFBBB6693D/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001342</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001342</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Approximation theory methods for solving elliptic eigenvalue problems</title><author><name sortKey="Still, G" sort="Still, G" uniqKey="Still G" first="G." last="Still">G. 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By applying results from complex approximation theory we obtain results on the approximation properties of special classes of solutions of Lu = 0 on G . These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a‐posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics (‘relaxed plasma’, ‘MHD‐equation’).</div></front></TEI><istex><corpusName>wiley</corpusName><author><json:item><name>G. Still</name><affiliations><json:string>University of Twente, Faculty of Mathematical Science, P.O. 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By applying results from complex approximation theory we obtain results on the approximation properties of special classes of solutions of Lu = 0 on G . These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a‐posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics (‘relaxed plasma’, ‘MHD‐equation’).</abstract><qualityIndicators><score>6.5</score><pdfVersion>1.3</pdfVersion><pdfPageSize>595 x 841 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractCharCount>802</abstractCharCount><pdfWordCount>7197</pdfWordCount><pdfCharCount>29806</pdfCharCount><pdfPageCount>11</pdfPageCount><abstractWordCount>125</abstractWordCount></qualityIndicators><title>Approximation theory methods for solving elliptic eigenvalue problems</title><refBibs><json:item><host><author></author><title>M. Abramowitz and I.A. Stegun, Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Thun, 1984).</title></host></json:item><json:item><host><volume>67</volume><pages><last>60</last><first>1</first></pages><author></author><title>Z. Angew. Math. Mech.</title></host></json:item><json:item><host><volume>67</volume><pages><first>7</first></pages><author></author><title>Nucl. Fusion</title></host></json:item><json:item><host><volume>67</volume><pages><last>680</last><first>654</first></pages><author></author><title>SIAM J. Numer. Anal.</title></host></json:item><json:item><host><volume>67</volume><pages><last>383</last><first>7</first></pages><author></author><title>Z. Angew. Math. Mech.</title></host></json:item><json:item><host><volume>67</volume><pages><last>102</last><first>89</first></pages><author></author><title>SIAM J. Numer. Anal.</title></host></json:item><json:item><host><volume>67</volume><pages><last>203</last><first>169</first></pages><author></author><title>Z. Angew. Math. Phys.</title></host></json:item><json:item><host><author></author><title>P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985).</title></host></json:item><json:item><host><volume>67</volume><pages><last>597</last><first>12</first></pages><author></author><title>Z. Angew. Math. Mech.</title></host></json:item><json:item><host><author></author><title>J.R Kuttler and V.G. Sigillito, Estimating Eigenvalues with A‐Posteriori/Apriori Inequalities (Pitman, Boston, 1985).</title></host></json:item><json:item><host><volume>67</volume><pages><last>760</last><first>727</first></pages><author></author><title>J. Math. Mech.</title></host></json:item><json:item><host><volume>67</volume><pages><last>122</last><first>31</first></pages><author></author><title>Usp. Mat. Nauk</title></host></json:item><json:item><host><volume>67</volume><pages><last>223</last><first>201</first></pages><author></author><title>Numer. Math.</title></host></json:item><json:item><host><author></author><title>G. Still, Defektminimierungsmethoden zur Lösung elliptischer Rand‐ und Eigenwertaufgaben, Habilitation, University of Trier (1989).</title></host></json:item><json:item><host><volume>67</volume><pages><last>290</last><first>7</first></pages><author></author><title>Z. Angew. Math. 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KGaA, Weinheim</copyright><eventGroup><event type="manuscriptReceived" date="2001-01-11"></event><event type="manuscriptRevised" date="2002-08-15"></event><event type="manuscriptAccepted" date="2002-09-30"></event><event type="firstOnline" date="2003-06-18"></event><event type="publishedOnlineFinalForm" date="2003-06-18"></event><event type="xmlConverted" agent="Converter:JWSART34_TO_WML3G version:2.3.2 mode:FullText source:HeaderRef result:HeaderRef" date="2010-03-02"></event><event type="xmlConverted" agent="Converter:WILEY_ML3G_TO_WILEY_ML3GV2 version:3.8.8" date="2014-02-11"></event><event type="xmlConverted" agent="Converter:WML3G_To_WML3G version:4.1.7 mode:FullText,remove_FC" date="2014-11-04"></event></eventGroup><numberingGroup><numbering type="pageFirst">468</numbering><numbering type="pageLast">478</numbering></numberingGroup><linkGroup><link type="toTypesetVersion" href="file:ZAMM.ZAMM468.pdf"></link></linkGroup></publicationMeta><contentMeta><countGroup><count type="figureTotal" number="2"></count><count type="tableTotal" number="5"></count><count type="referenceTotal" number="16"></count></countGroup><titleGroup><title type="main" xml:lang="en">Approximation theory methods for solving elliptic eigenvalue problems</title><title type="short" xml:lang="en">Approximation for elliptic eigenvalue problems</title></titleGroup><creators><creator xml:id="au1" creatorRole="author" affiliationRef="#a1"><personName><givenNames>G.</givenNames><familyName>Still</familyName></personName><contactDetails><email>g.j.still@math.utwente.nl</email></contactDetails></creator></creators><affiliationGroup><affiliation xml:id="a1" countryCode="NL" type="organization"><unparsedAffiliation>University of Twente, Faculty of Mathematical Science, P.O. 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Singular trial functions are applied to smooth the problem at corner points of <i>G</i> . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. 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Box 217, 7500 AE Enschede, The Netherlands</affiliation><affiliation>E-mail: g.j.still@math.utwente.nl</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="article" displayLabel="article"></genre><originInfo><publisher>WILEY‐VCH Verlag</publisher><place><placeTerm type="text">Berlin</placeTerm></place><dateIssued encoding="w3cdtf">2003-07-07</dateIssued><dateCaptured encoding="w3cdtf">2001-01-11</dateCaptured><dateValid encoding="w3cdtf">2002-09-30</dateValid><copyrightDate encoding="w3cdtf">2003</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType><extent unit="figures">2</extent><extent unit="tables">5</extent><extent unit="references">16</extent></physicalDescription><abstract lang="en">Eigenvalue problems with elliptic operators L on a domain G ⊂ R2 are considered. By applying results from complex approximation theory we obtain results on the approximation properties of special classes of solutions of Lu = 0 on G . These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a‐posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics (‘relaxed plasma’, ‘MHD‐equation’).</abstract><subject lang="en"><genre>keywords</genre><topic>elliptic eigenvalue problems</topic><topic>complex approximation</topic><topic>degree of approximation</topic><topic>a‐posteriori error bounds</topic><topic>defect‐minimization method</topic></subject><relatedItem type="host"><titleInfo><title>ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik</title><subTitle>Applied Mathematics and Mechanics</subTitle></titleInfo><titleInfo type="abbreviated"><title>Z. angew. Math. 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