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Domain decomposition methods with overlapping subdomains for the time-dependent problems of mathematical physics. (English) Zbl 1156.65084

Summary: 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 time-dependent problems with synchronous and asynchronous implementations has been investigated.
For nonstationary problems with self-adjoint operators, we consider three different types of decomposition operators corresponding to the Dirichlet and Neumann conditions on the subdomain boundaries. General stability conditions are obtained for the region-additive schemes. We focus on the accuracy of domain decomposition schemes. In particular, the dependence of the convergence rate on the width of subdomain overlapping is investigated as the primary property.
New classes of domain decomposition schemes for nonstationary problems, based on the subdomain overlapping and minimal data exchange in solving problems in subdomains, are constructed.

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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