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A Localized Version of the Method of Fundamental Solutions in a Multi-level Context

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Titel: A Localized Version of the Method of Fundamental Solutions in a Multi-level Context
Autoren: Gáspár, Csaba
Quelle: Periodica Polytechnica Civil Engineering; Vol. 67 No. 3 (2023); 716-724 ; 1587-3773 ; 0553-6626
Verlagsinformationen: Budapest University of Technology and Economics
Publikationsjahr: 2023
Bestand: Periodica Polytechnica (Budapest University of Technology and Economics)
Schlagwörter: method of fundamental solutions, local schemes, multi-level methods, quadtrees
Beschreibung: The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.
Publikationsart: article in journal/newspaper
Dateibeschreibung: application/pdf
Sprache: English
Relation: https://pp.bme.hu/ci/article/view/21535/9732; https://pp.bme.hu/ci/article/view/21535
Verfügbarkeit: https://pp.bme.hu/ci/article/view/21535
Rights: Copyright (c) 2023 Periodica Polytechnica Civil Engineering
Dokumentencode: edsbas.180D4009
Datenbank: BASE
Beschreibung
Abstract:The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.