Contemporary Mathematics Volume 157, 1994 Domain Decomposition Methods using Modified Basis Functions V.I. AGOSHKOV ABSTRACT. The paper investigates domain decomposition algorithms based on the use of modified basis functions in projective-difference approximation of problems. We use usual basis functions with support diameters of the order of the mesh size inside subdomains ([1-5],(9-11)}. On subdomain interface we consider instead basis functions with supports of greater size, which are extended to subdomains in a harmonic manner. We consider two boundary-value problems. The first of them is the Dirichlet problem in a rectangle n, the second one is a problem with natural boundary conditions in a "complicated" domain fi C S1, included into S1. In the Dirichlet problem we introduce modified basis functions and construct a "simple" precon- ditioner. Then we use the same modified basis functions to solve the problem with natural boundary conditions. We investigate the properties of the matri- ces arisen in Galerkin's approximations and present the convergence results of domain decomposition algorithms based on iterative processes of minimal cor- rections and locally optimal three-steps methods. We show the results of some numerical experiments. In the paper real-valued functions and well-known functional spaces L2(S1), W:l(S1), W:}(n)' Wi(S1), c1(n) are used. 1. Statement of Problem Let us consider the following problem with forced boundary conditions ( u = 0 on 8S1): find u(x11 x2) E W:} (S1) such that the relationship (1.1) a(u,v) = (f,v) 1991 Mathematics Subject Classification 65N55. Supported by "Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia" This paper is in final form and no version of it will be submitted for publication elsewhere 3 © 1994 American Mathematical Society 0271-4132/94 $1.00 + S.25 per page
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