Domain decomposition (DD) has been widely used to design parallel efficient algorithms for solving elliptic problems. In this thesis, we focus on improving the efficiency of DD methods and applying them to more general problems. Specifically, we propose efficient variants of the vertex space DD method and minimize the complexity of general DD methods. In addition, we apply DD algorithms to coupled elliptic systems, singular Neumann boundary problems and linear algebraic systems.
We successfully improve the vertex space DD method of Smith by replacing the exact edge, vertex dense matrices by approximate sparse matrices. It is extremely expensive to calculate, invert and store the exact vertex and edge Schur complement dense sub-matrices in the vertex space DD algorithm. We propose several approximations for these dense matrices, by using Fourier approximation and an algebraic probing technique. Our numerical and theoretical results show that these variants retain the fast convergence rate and greatly reduce the computational cost.
We develop a simple way to reduce the overall complexity of domain decomposition methods through choosing the coarse grid size. For sub-domain solvers with different complexities, we derive the optimal coarse grid size Hopt, which asymptotically minimizes the total computational cost of DD methods under the sequential and parallel environments. The overall complexity of DD methods is significantly reduced by using this optimal coarse grid size.
We apply the additive and multiplicative Schwarz algorithms to solving coupled elliptic systems. Using the Dryja-Widlund framework, we prove that their convergence rates are independent of both the mesh and the coupling parameters. We also construct several approximate interface sparse matrices by using Sobolev inequalities, Fourier analysis and probe technique.
We further discuss the application of DD to the singular Neumann boundary value problems. We extend the general framework to these problems and show how to deal with the null space in practice. Numerical and theoretical results show that these modified DD methods still have optimal convergence rate.
By using the DD methodology, we propose algebraic additive and multiplicative Schwarz methods to solve general sparse linear algebraic systems. We analyze the eigenvalue distribution of the iterative matrix of each each algebraic DD method to study the convergence behavior.