Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i>
We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></s...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-02-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/17/3/100 |
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| author | Andrea Adriani Stefano Serra-Capizzano Cristina Tablino-Possio |
| author_facet | Andrea Adriani Stefano Serra-Capizzano Cristina Tablino-Possio |
| author_sort | Andrea Adriani |
| collection | DOAJ |
| description | We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> preconditioning when the variable coefficient wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented. |
| first_indexed | 2024-04-24T18:37:54Z |
| format | Article |
| id | doaj.art-982beee8e37c446a8cce991706735d47 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-04-24T18:37:54Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-982beee8e37c446a8cce991706735d472024-03-27T13:17:14ZengMDPI AGAlgorithms1999-48932024-02-0117310010.3390/a17030100Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i>Andrea Adriani0Stefano Serra-Capizzano1Cristina Tablino-Possio2Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, ItalyDepartment of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, ItalyDepartment of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, ItalyWe consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> preconditioning when the variable coefficient wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.https://www.mdpi.com/1999-4893/17/3/100Caputo fractional derivativesHelmholtz equationseigenvalue asymptotic distributionspectral symbolclusteringGeneralized Locally Toeplitz sequences |
| spellingShingle | Andrea Adriani Stefano Serra-Capizzano Cristina Tablino-Possio Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> Algorithms Caputo fractional derivatives Helmholtz equations eigenvalue asymptotic distribution spectral symbol clustering Generalized Locally Toeplitz sequences |
| title | Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> |
| title_full | Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> |
| title_fullStr | Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> |
| title_full_unstemmed | Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> |
| title_short | Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number <i>μ</i> |
| title_sort | clustering distribution analysis and preconditioned krylov solvers for the approximated helmholtz equation and fractional laplacian in the case of complex valued unbounded variable coefficient wave number i μ i |
| topic | Caputo fractional derivatives Helmholtz equations eigenvalue asymptotic distribution spectral symbol clustering Generalized Locally Toeplitz sequences |
| url | https://www.mdpi.com/1999-4893/17/3/100 |
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