6 . Operator Theoretical Analysis to Domain Decomposition Methods

The purpose of the present paper is to give a brief summary of our recent study on the domain decomposition method from an operator theoretical point of view. There are a large number of works devoted to the mathematical analysis of the domain decomposition methods. Most of these works carry out the convergence analysis without any assumptions of general nature on the geometry of the decomposition. However, from the viewpoint of mathematical theory as well as from that of applications in science and engineering, we are seriously interested in the effect of relationships between the rate of convergence of iterations and the geometric shape of decomposed domains. Moreover, the choice of relaxation parameters is of importance. Our method enables us to get explicit convergence factors under some assumptions on geometric shapes of decomposed domains. Furthermore, our convergence theorems give information on the choice of relaxation parameters which guarantees a fast convergence. The problem considered in this paper is well discussed in the monograph by A. Quarteroni and A. Valli (Domain Decomposition Methods for Partial Differential Equations, Oxford, 1999), and the results described here may be said to be particular cases of theorems presented in their monograph. However, the advantage of employing our method is already described above. We shall present a rough sketch of the method of analysis and theorems without the proofs; for the complete proofs, we refer to [Fuj97], [FKKN96], [FFS98] and [FS97].