Defect correction and domain decomposition for second-order boundary value problems
Highly accurate approximation is obtained through the techniques of defect correction and domain decomposition for second-order elliptic boundary value problems on a disc. The basic solution is computed using the Schwarz domain decomposition procedure and bilinear Galerkin finite element approximati...
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| Veröffentlicht in: | Journal of computational and applied mathematics Jg. 130; H. 1; S. 41 - 51 |
|---|---|
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
Elsevier B.V
01.05.2001
Elsevier |
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | Highly accurate approximation is obtained through the techniques of
defect correction and domain decomposition for second-order elliptic boundary value problems on a disc. The basic solution is computed using the Schwarz domain decomposition procedure and bilinear Galerkin finite element approximation on each subdomain to get an O(
h
2) accurate basic solution in higher-order discrete Sobolev norms. The defects are then computed using high-order polynomials (Lagrange polynomials or splines) to get as many O(
h
2) corrections as possible. |
|---|---|
| AbstractList | Highly accurate approximation is obtained through the techniques of
defect correction and domain decomposition for second-order elliptic boundary value problems on a disc. The basic solution is computed using the Schwarz domain decomposition procedure and bilinear Galerkin finite element approximation on each subdomain to get an O(
h
2) accurate basic solution in higher-order discrete Sobolev norms. The defects are then computed using high-order polynomials (Lagrange polynomials or splines) to get as many O(
h
2) corrections as possible. |
| Author | Chibi, Ahmed-Salah |
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| Cites_doi | 10.1137/0723026 10.1090/S0025-5718-1974-0373325-9 10.1007/BF01432879 10.1007/BF01394455 10.1002/num.1690080506 10.1093/imanum/8.2.149 |
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| Keywords | Bilinear finite elements Defect correction 65J10 Elliptic problems 65B05 65N55 Schwarz domain decomposition 65N30 Second order Bilinear system Partial differential equation Finite element method Boundary value problem Schwarz mzthod Elliptic problem Lagrange interpolation Defect Galerkin method Domain decomposition Corrections Finite element Bilinear approximation |
| Language | English |
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| References | Moore (BIB7) 1988; 52 Stetter (BIB11) 1978; 29 P.L. Lions, On the Schwarz alternating method I, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Perieux (Eds.), Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 1–41. A.S. Chibi, Defect correction and Galerkin's method for second order elliptic boundary value problems, Ph.D. Thesis, Imperial College, London, 1989. Nitsche, Schatz (BIB8) 1974; 26 Skeel (BIB9) 1986; 23 M. Boulbrachène, P. Courtey-Dumont, J.C. Miellou, Mixing finite element and finite differences in a subdomain method, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Perieux (Eds.), Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 198–216. Böhmer, Stetter (Eds.) (BIB2) 1984 Stariüs (BIB10) 1974; 28 Barrett, Moore, Morton (BIB1) 1988; 8 Chibi, Moore (BIB5) 1992; 8 Böhmer (10.1016/S0377-0427(99)00392-1_BIB2) 1984 Stetter (10.1016/S0377-0427(99)00392-1_BIB11) 1978; 29 Nitsche (10.1016/S0377-0427(99)00392-1_BIB8) 1974; 26 Barrett (10.1016/S0377-0427(99)00392-1_BIB1) 1988; 8 10.1016/S0377-0427(99)00392-1_BIB4 Chibi (10.1016/S0377-0427(99)00392-1_BIB5) 1992; 8 Stariüs (10.1016/S0377-0427(99)00392-1_BIB10) 1974; 28 10.1016/S0377-0427(99)00392-1_BIB3 10.1016/S0377-0427(99)00392-1_BIB6 Skeel (10.1016/S0377-0427(99)00392-1_BIB9) 1986; 23 Moore (10.1016/S0377-0427(99)00392-1_BIB7) 1988; 52 |
| References_xml | – volume: 52 start-page: 565 year: 1988 end-page: 582 ident: BIB7 publication-title: Defect correction from a Galerkin viewpoint, Numer. Math. – volume: 26 start-page: 937 year: 1974 end-page: 958 ident: BIB8 article-title: Interior estimates for Ritz Galerkin Methods publication-title: Math. Comput. – reference: M. Boulbrachène, P. Courtey-Dumont, J.C. Miellou, Mixing finite element and finite differences in a subdomain method, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Perieux (Eds.), Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 198–216. – volume: 8 start-page: 149 year: 1988 end-page: 184 ident: BIB1 article-title: Optimal recovery in the finite element method, Part 1: recovery from weighted publication-title: IMA J. Numer. Anal. – reference: A.S. Chibi, Defect correction and Galerkin's method for second order elliptic boundary value problems, Ph.D. Thesis, Imperial College, London, 1989. – reference: P.L. Lions, On the Schwarz alternating method I, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Perieux (Eds.), Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 1–41. – volume: 28 start-page: 243 year: 1974 end-page: 258 ident: BIB10 article-title: Composite mesh difference methods for elliptic problems publication-title: Numer. Math. – volume: 8 start-page: 469 year: 1992 end-page: 491 ident: BIB5 article-title: Defect correction and Galerkin's method for second order boundary value problems publication-title: J. Numer. Methods for PDEs – year: 1984 ident: BIB2 publication-title: Defect Correction Methods Theory and Applications – volume: 29 start-page: 425 year: 1978 end-page: 443 ident: BIB11 article-title: The defect correction principle and discretisation methods publication-title: Numer. Math. – volume: 23 start-page: 393 year: 1986 end-page: 402 ident: BIB9 article-title: The order of accuracy for deferred correction using uncentred end formulae publication-title: SIAM J. Numer. Anal. – year: 1984 ident: 10.1016/S0377-0427(99)00392-1_BIB2 – volume: 23 start-page: 393 year: 1986 ident: 10.1016/S0377-0427(99)00392-1_BIB9 article-title: The order of accuracy for deferred correction using uncentred end formulae publication-title: SIAM J. Numer. Anal. doi: 10.1137/0723026 – ident: 10.1016/S0377-0427(99)00392-1_BIB3 – ident: 10.1016/S0377-0427(99)00392-1_BIB4 – volume: 26 start-page: 937 issue: 128 year: 1974 ident: 10.1016/S0377-0427(99)00392-1_BIB8 article-title: Interior estimates for Ritz Galerkin Methods publication-title: Math. Comput. doi: 10.1090/S0025-5718-1974-0373325-9 – volume: 29 start-page: 425 year: 1978 ident: 10.1016/S0377-0427(99)00392-1_BIB11 article-title: The defect correction principle and discretisation methods publication-title: Numer. Math. doi: 10.1007/BF01432879 – volume: 28 start-page: 243 year: 1974 ident: 10.1016/S0377-0427(99)00392-1_BIB10 article-title: Composite mesh difference methods for elliptic problems publication-title: Numer. Math. doi: 10.1007/BF01394455 – volume: 8 start-page: 469 year: 1992 ident: 10.1016/S0377-0427(99)00392-1_BIB5 article-title: Defect correction and Galerkin's method for second order boundary value problems publication-title: J. Numer. Methods for PDEs doi: 10.1002/num.1690080506 – ident: 10.1016/S0377-0427(99)00392-1_BIB6 – volume: 52 start-page: 565 year: 1988 ident: 10.1016/S0377-0427(99)00392-1_BIB7 publication-title: Defect correction from a Galerkin viewpoint, Numer. Math. – volume: 8 start-page: 149 year: 1988 ident: 10.1016/S0377-0427(99)00392-1_BIB1 article-title: Optimal recovery in the finite element method, Part 1: recovery from weighted L2 fits publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/8.2.149 |
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defect correction and domain decomposition for second-order elliptic boundary value... |
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| SubjectTerms | Acceleration of convergence Bilinear finite elements Defect correction Elliptic problems Exact sciences and technology Mathematics Numerical analysis Numerical analysis in abstract spaces Numerical analysis. Scientific computation Partial differential equations, boundary value problems Schwarz domain decomposition Sciences and techniques of general use |
| Title | Defect correction and domain decomposition for second-order boundary value problems |
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