High-Order Solution of Elliptic Partial Differential Equations in Domains Containing Conical Singularities

Author: Li, Zhiyi

Year: 2009

Degree: Dissertation (Ph.D.)

Advisor: Bruno, Oscar P.

Committee Members: Bruno, Oscar P.; Shepherd, Joseph E.; Schroeder, Peter; Hou, Thomas Y.

Option: Applied And Computational Mathematics

DOI: 10.7907/VEEB-AV75

Abstract

In this thesis we introduce an algorithm, based on the boundary integral equation method, for the numerical evaluation of singular solutions of the Laplace equation in three dimensional space, with singularities induced by a conical point on an otherwise smooth boundary surface. This is a model version of a fundamental problem in science and engineering: accurate evaluation of solutions of Partial Differential Equations in domains whose boundaries contain geometric singularities. For simplicity we assume a small region near the conical point coincides with a straight cone of given cross section; otherwise the boundary surface is not restricted in any way. Our numerical results demonstrate excellent convergence as discretizations are refined, even at the singular point where the solutions tend to infinity.

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