A domain decomposition method for linear exterior boundary value problems
In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral eq...
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| Published in: | Applied mathematics letters Vol. 11; no. 6; pp. 1 - 9 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Elsevier Ltd
01.11.1998
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| ISSN: | 0893-9659, 1873-5452 |
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| Abstract | In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided. |
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| AbstractList | In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided. |
| Author | Hernandez, E.C. Gatica, G.N. Mellado, M.E. |
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| Cites_doi | 10.1006/jmaa.1995.1029 10.1016/0022-247X(77)90186-X 10.1007/BF01398917 10.1137/0731036 |
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| Keywords | Steklov-Poincaré operator Dirichlet-Robin sweep Iteration by subdomains Dirichlet-to-Neumann mapping |
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| References | Le Tallec (BIB1) 1994; 1 Gatica, Hsiao (BIB5) 1995; 189 Gatica, Mellado (BIB9) 1998 Hsiao, Zhang (BIB6) 1994; 31 Hsiao, Khoromskij, Wendland (BIB3) 1994 Hsiao, Wendland (BIB7) 1977; 58 Marini, Quarteroni (BIB8) 1989; 55 M.E. Mellado, Domain decomposition methods for exterior problems in potential theory and elasticity, Ph.D. Thesis, Universidad de Concepción, (in preparation). Quarteroni, Valli (BIB2) 1991 Langer (BIB4) 1994; 157 Marini (10.1016/S0893-9659(98)00093-7_BIB8) 1989; 55 Le Tallec (10.1016/S0893-9659(98)00093-7_BIB1) 1994; 1 10.1016/S0893-9659(98)00093-7_BIB10 Quarteroni (10.1016/S0893-9659(98)00093-7_BIB2) 1991 Hsiao (10.1016/S0893-9659(98)00093-7_BIB7) 1977; 58 Gatica (10.1016/S0893-9659(98)00093-7_BIB9) 1998 Gatica (10.1016/S0893-9659(98)00093-7_BIB5) 1995; 189 Hsiao (10.1016/S0893-9659(98)00093-7_BIB6) 1994; 31 Hsiao (10.1016/S0893-9659(98)00093-7_BIB3) 1994 Langer (10.1016/S0893-9659(98)00093-7_BIB4) 1994; 157 |
| References_xml | – volume: 55 start-page: 575 year: 1989 end-page: 598 ident: BIB8 article-title: A relaxation procedure for domain decomposition methods using finite elements publication-title: Numerische Mathematik – reference: M.E. Mellado, Domain decomposition methods for exterior problems in potential theory and elasticity, Ph.D. Thesis, Universidad de Concepción, (in preparation). – year: 1994 ident: BIB3 article-title: Boundary integral operators and domain decomposition – volume: 157 start-page: 335 year: 1994 end-page: 344 ident: BIB4 article-title: Parallel iterative solution of symmetric coupled FE/BE-equations via domain decomposition publication-title: Domain Decomposition Methods in Science and Engineering – volume: 189 start-page: 442 year: 1995 end-page: 461 ident: BIB5 article-title: The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems publication-title: Journal of Mathematical Analysis and Applications – volume: 31 start-page: 680 year: 1994 end-page: 694 ident: BIB6 article-title: Optimal order multigrid methods for solving exterior boundary value problems publication-title: SIAM Journal on Numerical Analysis – year: 1998 ident: BIB9 article-title: Nonoverlapping domain decomposition methods for linear and nonlinear exterior boundary value problems publication-title: Technical Report 98-02 – start-page: 179 year: 1991 end-page: 203 ident: BIB2 article-title: Theory and application of Steklov-Poincaré operators for boundary value problems publication-title: Applied and Industrial Mathematics – volume: 58 start-page: 449 year: 1977 end-page: 481 ident: BIB7 article-title: A finite element method for some integral equations of the first kind publication-title: Journal of Mathematical Analysis and Applications – volume: 1 start-page: 121 year: 1994 end-page: 220 ident: BIB1 article-title: Domain decomposition methods in computational mechanics publication-title: Computational Mechanics Advances – year: 1994 ident: 10.1016/S0893-9659(98)00093-7_BIB3 article-title: Boundary integral operators and domain decomposition – ident: 10.1016/S0893-9659(98)00093-7_BIB10 – volume: 189 start-page: 442 year: 1995 ident: 10.1016/S0893-9659(98)00093-7_BIB5 article-title: The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems publication-title: Journal of Mathematical Analysis and Applications doi: 10.1006/jmaa.1995.1029 – year: 1998 ident: 10.1016/S0893-9659(98)00093-7_BIB9 article-title: Nonoverlapping domain decomposition methods for linear and nonlinear exterior boundary value problems – volume: 58 start-page: 449 year: 1977 ident: 10.1016/S0893-9659(98)00093-7_BIB7 article-title: A finite element method for some integral equations of the first kind publication-title: Journal of Mathematical Analysis and Applications doi: 10.1016/0022-247X(77)90186-X – start-page: 179 year: 1991 ident: 10.1016/S0893-9659(98)00093-7_BIB2 article-title: Theory and application of Steklov-Poincaré operators for boundary value problems – volume: 1 start-page: 121 year: 1994 ident: 10.1016/S0893-9659(98)00093-7_BIB1 article-title: Domain decomposition methods in computational mechanics publication-title: Computational Mechanics Advances – volume: 55 start-page: 575 year: 1989 ident: 10.1016/S0893-9659(98)00093-7_BIB8 article-title: A relaxation procedure for domain decomposition methods using finite elements publication-title: Numerische Mathematik doi: 10.1007/BF01398917 – volume: 31 start-page: 680 year: 1994 ident: 10.1016/S0893-9659(98)00093-7_BIB6 article-title: Optimal order multigrid methods for solving exterior boundary value problems publication-title: SIAM Journal on Numerical Analysis doi: 10.1137/0731036 – volume: 157 start-page: 335 year: 1994 ident: 10.1016/S0893-9659(98)00093-7_BIB4 article-title: Parallel iterative solution of symmetric coupled FE/BE-equations via domain decomposition |
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| Title | A domain decomposition method for linear exterior boundary value problems |
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