"The alternating Schwarz method: mathematical foundation and parallel implementation"The alternating Schwarz method belongs to the class of domain decomposition methods for solving partial differential equations. The given domain is partitioned into a number of subregions. The original problem can be reformulated as a family of subproblems of reduced size defined on the subdomains. Based on solving these subproblems, a preconditioner is constructed for the linear system which evolves from the discretization of the original problem. Using this preconditioner, the solution for the linear system is obtained with a preconditioned conjugate gradient method. One of the major advantages of domain decomposition methods is the natural parallelism in solving the subproblems. The purpose of this thesis is to discuss the mathematical aspects of the alternating Schwarz method, and to give insight into its implementation on a distributed memory machine. Chapter 1 provides an introduction to the finite element method. The classical and weak formulation, and the
finite element approximation of the Poisson Problem are discussed. An algorithm is obtained which is used to discretize the problem, and to solve the linear system with the conjugate gradient method. In chapter 2 the mathematical foundation of the alternating Schwarz method is considered. It begins with the classical formulation of the method. Focussing on the additive Schwarz method, its variational formulation and the characterization in terms of projection operators is derived. On the finite dimensional level, the construction of the additive Schwarz preconditioner is presented, and the resulting preconditioned conjugate gradient method is introduced. Chapter 3 is dedicated to a discussion of the parallel implementation of the additive Schwarz method. The characterization of a distributed memory machine is given, and the basics of the message passing system MPI are introduced. Then, it is shown how the discretization of the problem and the process of obtaining the system of linear equations can be carried out in parallel. The preconditioned conjugate gradient method is parallelized, and a compact form of the parallel algorithm is given. Finally, the results of some numerical experiments are presented.