A Study of Multigrid Methods with Application to Convection-Diffusion Problems

Abstract

The work accomplished in this dissertation is concerned with the numerical solution of linear elliptic partial differential equations in two dimensions, in particular modeling diffusion and convection-diffusion. The finite element method applied to this type of problem gives rise to a linear system of equations of the form Ax=b. It is well known that direct and classical iterative (or relaxation) methods can be used to solve such systems of equations, but the efficiency deteriorates in the limit of a highly refined grid. The multigrid methodologies discussed in this dissertation evolved from attempts to correct the limitations of the conventional solution methods. The aim of the project is to investigate the performance of multigrid methods for diffusion problems, and to explore the potential of multigrid in cases where convection dominates.