領域分割法の収束に関する解析的ならびに数値的研究,I Analytical and numerical study of convergence of the domain decomposition method, I
In this paper and forthcoming ones, we report the result of our recent study on the rate of convergence of iterations met in the domain decomposition method (DDM) which is now extensively applied in order to solve boundary value problems for partial differential equations. Our study has been carried out by an operator-theoretic analysis as well as by numerical experiments, and is focused on derivation of clear-cut estimates of the decay rate of errors and on clarification of the influence of the way of domain-decomposition. In the present paper, we consider for simplicity the boundary value problem for the Poisson equation in a bounded domain Ω in R^2 which we divide by an artificial dividing curve γ into two sub-domains Ω_1 and Ω_2. As for the iteration algorithm, we deal here only with the so-called Dirichlet-Neumann iteration, of which the unit cycle is composed of 1) solving the sub-problem in Ω_1 with the current approximation λ^<(k)> on γ as the Dirichlet data, 2) then solving the sub-problem in Ω_2 with the Neumann data on γ which are calculated from the result of the preceding sub-problem 1, and 3) finally, renewing λ^<(k)> to λ^<(k+1)> with the relaxation parameter θ. On the other hand, our assumptions on the geometry of the decomposition are described in terms of the linear or circular reflection of sub-domains with respect to γ and in terms of dilation of sub-domains combined with the reflection mentioned above. Under such assumptions, we derive estimates of the exponential decay rate of the error for each θ, and present the optimal choice of θ together with the resulting optimal rate of convergence. Although the method of our analysis is based on the theory which was invented by the first author in use of self-adjoint operators derived from the Steklov-Poincare operators, we have re-organized it anew to pave the way toward generalizations in various directions. The here reported numerical experiments implemented by the finite difference method are confirming the validity, the significance and the limitation of the theoretical result. As a matter of fact, they also have played in the course of our study the role to give a pilot-light into promising problems of further interests and for further consideration.