UBC Theses and Dissertations
A numerical investigation of two boundary element methods Quek, Mui Hoon
This thesis investigates the viability of two boundary element methods for solving steady state problems, the continuous least squares method and the Galerkin minimization technique. In conventional boundary element methods, the singularities of the fundamental solution involved are usually located at fixed points on the boundary of the problem's domain or on an auxiliary boundary. This leads to some difficulties: when the singularities are located on the problem domain's boundary, it is not easy to evaluate the solution for points on or near that boundary whereas if the singularities are placed on an auxiliary boundary, this auxiliary boundary would have to be carefully chosen. Hence the methods studied here allow the singularities, initially located at some auxiliary boundary, to move until the best positions are found. These positions are determined by attempting to minimize the error via the least squares or the Galerkin technique. This results in a highly accurate, adaptive, but nonlinear method. We study various methods for solving systems of nonlinear equations resulting from the Galerkin technique. A hybrid method has been implemented, which involves the objective function from the least squares method while the gradient is due to the Galerkin method. Numerical examples involving Laplace's equation in two dimensions are presented and results using the discrete least squares method, the continuous least squares method and the Galerkin method are compared and discussed. The continuous least squares method appears to give the best results for the sample problems tried.
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