Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients
| Autor: | Françoise Willien, E. Flauraud, Frédéric Nataf |
|---|---|
| Přispěvatelé: | Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Ruprecht, Liliane |
| Rok vydání: | 2006 |
| Předmět: |
Mathematical optimization
MathematicsofComputing_NUMERICALANALYSIS 010103 numerical & computational mathematics Positive-definite matrix computer.software_genre 01 natural sciences Deflation Matrix (mathematics) Multigrid method Optimized interface conditions Discontinuous and anisotropic coefficients Applied mathematics Symmetric matrix Domain decomposition method 0101 mathematics Eigenvalues and eigenvectors Mathematics Numerical linear algebra Preconditioner Applied Mathematics Domain decomposition methods [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA] Porous media flow 010101 applied mathematics Computational Mathematics computer [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| Zdroj: | Journal of Computational and Applied Mathematics Journal of Computational and Applied Mathematics, 2006, 189, pp.539-554 Journal of Computational and Applied Mathematics, Elsevier, 2006, 189, pp.539-554 |
| ISSN: | 0377-0427 |
| DOI: | 10.1016/j.cam.2005.05.019 |
| Popis: | When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the “bad” eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis’ result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches. |
| Databáze: | OpenAIRE |
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