Algebraic Theory of Schwarz Methods for Domain Decomposition
by
Daniel B Szyld
Temple University, Philadelphia, USA
Coauthors: Michele Benzi (Emory U., Atlanta), Andreas Frommer (Wuppertal), Reinhard Nabben (Bielefeld)

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. Similarly, convergence proofs are given for restrictive additive and multiplicative Schwarz methods, for which no convergence theory was available before. References: M. Benzi, A. Frommer, R. Nabben, and D. B. Szyld, Algebraic theory of multiplicative Schwarz methods. Numerische Mathematik, to appear in 2001. A. Frommer and D. B. Szyld. Weighted max norms, splittings, and overlapping additive Schwarz iterations. Numerische Mathematik, 83:259--278, 1999. A. Frommer and D. B. Szyld. An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms. SIAM Journal on Numerical Analysis, 39:463-479, 2001. R. Nabben and D. B. Szyld. Convergence theory of restricted multiplicative Schwarz methods. Research Report, May 2001.

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Date received: February 14, 2001