Analysis of Preconditioners for Domain Decomposition.
YALE UNIV NEW HAVEN CT DEPT OF COMPUTER SCIENCE
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Domain decomposition is a class of techniques that are designed to solve elliptic problems on irregular domains and on multiprocessor systems. Typically, a domain is decomposed into many smaller regular subdomains and the capacitance system governing the interface unknowns is solved by some version of the preconditioned conjugate gradient method. In this paper, we show that for a simple model problem -- Poissons equation on a rectangle decomposed into two smaller rectangles -- the capacitance system can be inverted exactly by fast Fourier transform. No iteration is needed. An exact eigen-decomposition of the capacitance matrix also makes possible a comparison of various preconditioners that have been proposed in the literature. For example, we show that in the limit as the aspect ratio of the two rectangles tend to infinity, the preconditioner proposed by Golub-Mayers becomes exact, but the one proposed by Dryja does not. Both preconditioners, however, are poor when the aspect ratio is small. Author
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