ISSN:
1572-9265
Keywords:
convergence
;
multilevel additive methods
;
unstructured meshes
;
65N30
;
65F10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract We 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 nonselfadjoint 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 on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2 L 2), whereρ = max1≤l≤L(h l +l− 1)/δ l,h l is the element size of thelth level mesh,δ l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02207701
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