Preconditioners and Fundamental Solutions

1. Introduction. The topic of this Licentiate thesis is the iterative solution of large, sparse systems of equations originating from discretizations of partial differential equations (PDEs). Our approach is to construct a preconditioner that is a truncated discrete convolution operator. The kernel is a fundamental solution, or an approximation of one. Our method generalizes to linear systems with block-Toeplitz structure without an underlying PDE. The problem of solving systems of equations is very important. It arises in many applications, both with and without the PDE connection. A large amount of time is spent in industrial codes to solve such systems. The research behind the thesis is presented in three papers, Paper A–C. Both Paper A and Paper C deal with iterative solution methods and preconditioning, and they explore two branches of the same idea. The latter paper includes analysis showing that the method we propose has favorable properties for a relevant model problem. Furthermore, numerical experiments verify that it can compete with an alternative, efficient preconditioning method for a more realistic fluid flow problem. In Paper B, we construct an algorithm for computing fundamental solutions of difference operators. The algorithm is used when constructing the preconditioner in Paper C. This summary is organized as follows: Section 2 reviews some elementary definitions and basic relations from the theory of PDE. Also, some relevant topics in numerical linear algebra (NLA) are reviewed. In section 3 we summarize the results from the three papers, and section 4 contains an overview of related pre-conditioners. Section 5 ends the summary with concluding remarks concerning future work and possible extensions of the method.