{"title":"片状光滑介质中𝑝有限元离散化的亥姆霍兹问题的波长显式稳定性和收敛性分析","authors":"M. Bernkopf, T. Chaumont-Frelet, J. Melenk","doi":"10.1090/mcom/3958","DOIUrl":null,"url":null,"abstract":"<p>We present a wavenumber-explicit convergence analysis of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h p\"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"33 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media\",\"authors\":\"M. Bernkopf, T. Chaumont-Frelet, J. Melenk\",\"doi\":\"10.1090/mcom/3958\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present a wavenumber-explicit convergence analysis of the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"h p\\\"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k\\\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.</p>\",\"PeriodicalId\":18456,\"journal\":{\"name\":\"Mathematics of Computation\",\"volume\":\"33 1\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-03-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics of Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3958\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3958","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
我们提出了对 h p hp 有限元方法的波数显式收敛性分析，该方法适用于一类在大波数 k k 下具有片断解析系数的异质亥姆霍兹问题。我们的分析涵盖了具有 Robin、精确 Dirichlet-to-Neumann、二阶吸收边界条件以及完全匹配层的异质 Helmholtz 方程。
Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media
We present a wavenumber-explicit convergence analysis of the hphp Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber kk. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.
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This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.
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