Analysis of Boundary Element Method and Finite Difference Method for the Laplace Equation

Abstract

The main intention of this study is to examine the boundary element method and finite difference method for finding an approximate solution of two dimensional Laplace equation. Since the two dimensional Laplace equation is considered, the boundaries of the domain are one dimensional and discretization is carried on the one dimensional boundary for the boundary element method and domain discretization is carried on for that of the finite difference method. A general description of both methods is explicated in this thesis. This explanation discusses on the types of element for the Boundary element method for the two dimensional Laplace equation. And in addition there is no detail idea on how to show stability consistency and convergence. So, we analyze stability, convergence and consistency of both presented methods and for this we apply theorems with some proofs and mainly focus on boundary element method. And also we compare the approximate solution of the two dimensional Laplace equation by these proposed methods based on test problems.