Domain Decomposition Methods with Overlapping Subdomains for the Time-Dependent Problems of Mathematical Physics

Abstract

At the present time, the domain decomposition methods are considered as the most promising ones for parallel computer systems. Nowadays success is attained mainly in solving approximately the classical boundary value problems for second-order elliptic equations. As for the time-dependent problems of mathematical physics, there are, in common use, approaches based on ordinary implicit schemes and implemented via iterative methods of the domain decomposition. An alternative technique is based on the non-iterative schemes (region-additive schemes). On the basis of the general theory of additive schemes a wide class of difference schemes (alternative directions, locally one-dimensional, factorized schemes, summarized approximation schemes, vector additive schemes, etc.) as applied to the domain decomposition technique for timedependent problems with synchronous and asynchronous implementations has been investigated. For nonstationary problems with self-adjoint operators, we have considered three different types of decomposition operators corresponding to the Dirichlet and Neumann conditions on the subdomain boundaries. General stability conditions have been obtained for the region-additive schemes. We focused on the accuracy of domain decomposition schemes. In particular, the dependence of the convergence rate on the width of subdomain overlapping has been investigated as the primary property. In the present paper, new classes of domain decomposition schemes for nonstationary problems, based on the subdomain overlaping and minimal data exchange in solving problems in subdomains, have been constructed. © 2008, Institute of Mathematics, NAS of Belarus. All rights reserved.