Domain Decomposition Methods For Elliptic Variational Problems

Domain decomposition is one of the most significant way for devising parallel algorithms that can benefit strongly from multiprocessor computers. Domain decomposition methods are generally based on the assumption that the given computational domain is partitioned into subdomains, which may or may not overlap. The alternative Schwarz method is undoubtedly the earliest example of domain decomposition methods. In this dissertation, we consider domain decomposition to solve elliptic variational problems, including variational inequalities, complementarity problem and partial differential equations.Complementarity problems are used to interpret and study the mathematical models of physics, mechanics, economics and optimal control, and various of equilibrium models that arise from traffic conveyance. The methods for the numerical solution of the complementarity problems are developed rapidly. At the present time, there are many iterative methods for solving complementarity problems. Domain decomposition method is a kind of pop iterative method studied by many researchers. For symmetric linear complementarity problems, most results are based on the assumption that the coefficient matrices are symmetric and positive definite or M matrices. In this dissertation, domain decomposition methods for the case of the coefficient matrices are symmetric and copositive are proposed. Convergence of the methods is established. And numerical results are present to show the efficiency of the methods.Domain decomposition methods for partial differential equations were developed in 1980s. From then on, the methods are recognized by more and more researchers. The Schwarz algorithms, which use Robin transmission conditions on the inner boundaries of the subdomains, is also called generalized Schwarz algorithms. Compared with the classical Schwarz algorithm, the generalized Schwarz algorithms replace transmission conditions on the interface between subdomains by the Robin conditions with parameters. In this dissertation, the convergence rate of a generalized additive Schwarz algorithm for solving boundary value problems of differential equations is studied. A quantitative analysis of the convergence rate is given for the one and two dimensional model Dirichlet problems. It shows that small overlapping is preferred for the generalized additive Schwarz algorithm. Some numerical tests also show that a greater acceleration of the algorithm can be obtained by choosing the parameter suitably. The alternative Schwarz method...