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Extending theory for domain decomposition algorithms to irregular subdomains. (English) Zbl 1140.65365

Langer, Ulrich (ed.) et al., Domain decomposition methods in science and engineering XVII. Selected papers based on the presentations at the 17th international conference on domain decomposition methods, St. Wolfgang/Strobl, Austria, July 3–7, 2006. Berlin: Springer (ISBN 978-3-540-75198-4/pbk). Lecture Notes in Computational Science and Engineering 60, 255-261 (2008).
From the introduction: In the theory of iterative substructuring domain decomposition methods, we typically assume that each subdomain is quite regular, e.g., the union of a small set of coarse triangles or tetrahedra. However, this is often unrealistic especially if the subdomains result from using a mesh partitioner. The subdomain boundaries might then not even be uniformly Lipschitz continuous. We note that existing theory establishes bounds on the convergence rate of the algorithms which are insensitive to even large jumps in the material properties across subdomain boundaries as reflected in the coefficients of the problem. The theory for overlapping Schwarz methods is less restrictive as far as the subdomain shapes are concerned, see e.g., but little has been known on the effect of large changes in the coefficients.
The purpose of this paper is to begin the development of a theory under much weaker assumptions on the partitioning. We will focus on a recently developed overlapping Schwarz method, which combines a coarse space adopted from an iterative substructuring method with local preconditioner components selected as in classical overlapping Schwarz methods, Le., based on solving problems on overlapping subdomains. This choice of the coarse component will allow us to prove results which are independent of coefficient jumps.
For the entire collection see [Zbl 1130.65004].

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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