Algebraic Theory of Additive and Multiplicative Schwarz
Daniel B. SzyldTemple University Philadelphia, Pennsylvania, USA.
July 19, 2000, 11.00 am, CERFACS Conference Room
Abstract :
The convergence of additive and multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular $M$-matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings.
The algebraic analysis presented complements the analysis usually done on these methods using Sobolev spaces. In some instances, such as in the proof of convergence of multiplicative Schwarz for nonsymmetric $M$-matrices, the hypotheses used in the algebraic analysis are much milder: no condition on the coarse grid correction is imposed, and in fact convergence is shown without the need for a coarse grid correction.
The new algebraic theory developed here is applied to a variety of situations:
the effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of coarse grid corrections (global coarse solves), and the convergence of different variants of restrictive additive Schwarz methods (RAS).
Joint work with Michele Benzi, Andreas Frommer, and Reinhard Nabben.
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