A convergence theory of multilevel additive Schwarz methods on unstructured meshes
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AbstractWe develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for non-selfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements oil the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the order O(rho(2)L(2)), where rho=max,(1 less than or equal to l less than or equal to L)(h(l)+h(l-1))/delta(l), h(l) is the element size of the lth level mesh, delta(l) the overlap of subdomains on the lth level mesh, and L the number of mesh levels.
All Author(s) ListChan TF, Zou J
Journal nameNumerical Algorithms
Volume Number13
Issue Number3-4
Pages365 - 398
LanguagesEnglish-United Kingdom
Keywordsconvergence; multilevel additive methods; unstructured meshes
Web of Science Subject CategoriesMathematics; Mathematics, Applied; MATHEMATICS, APPLIED

Last updated on 2020-03-04 at 00:57