Questions of domain decomposition are considered in connection with the numerical solution of parabolic problems in composite domains. Special schemes, which split the problem into subproblems to be solved in simple domains, are proposed.
Regionally-additive difference schemes in domains with and without overlapping are investigated for stability.
Decomposition methods are highly promising for solving problems of mathematical physics in composite domains.
The basic idea of these methods is to split the computation domain into several simpler subdomains, some of which may overlap. Such segmentation of the domain also implies segmentation of the entire computation algorithm into several weakly interdependent subproblems /i, 2/.
It also facilitates adaptation of the computation algorithm to the architecture of multiprocessor computers.
The greatest advances have been achieved in research into decomposition methods as applied to boundary-value problems for elliptic equations: the capacity matrix method, the Schwarz alternating method, iterative process in subspaces, and so On /3-5/.
In connection with decomposition methods for non-stationary problems major attention is being focussed on the formulation of exchange boundary conditions on the common boundaries of non-overlapping subdomains /i, 2/. One-dimensional problems for the heat-conduction equation were considered in /6/.
In /7, 8/, considering the same problems, the stability of difference schemes in relation to different exchange conditions was investigated. Two-dimensional heat-conduction problems were studied in /9/. The schemes considered there were applied to domains with and without overlapping, analogous to the classical alternating-direction schemes.
The topic of this paper is decomposition difference schemes for multidimensional parabolic problems.
We shall propose two types of difference schemes with domain decomposition for problems with and without overlapping.
In the first approach the basic idea is to use weighted schemes with a variable weight, depending on the point. The schemes of the second type are based on a symmetric decomposition of the space operator of the difference scheme.
The stability of the splitting schemes will be investigated. Their distinctive feature is that the problem is always solved in a simple subdomain. In the most general case one may be working with different computational grids in each subdomain. Other possible generalizations of our difference schemes will be indicated.