# CAIMS Prize Award

Department of Mathematics and Applied Mathematics, University of Toronto

Supervisor: G.F.D. Duff

Thesis title: Numerical Methods for Elliptic Boundary Value Problems with Singularities

Abstract:

Part 1: Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are first reported for the interface problems of the elliptic equation, Delta(u)+u=0, which yield extremely accurate solutions and surprisingly small condition numbers of the coefficient matrix with only a modest computational effort.

The advantage of the boundary methods over a standard finite difference or finite element method is that it can cope with complicated boundaries and boundary conditions as well as singularities and infinite domains. From a computational point of view, boundary methods are easy to use, benefiting from the reduced complexity of a boundary approximation. In addition it is often possible to control the error in the approximation by the computable error in the boundary, even for the elliptic problems which do not possess a maximum principle.

Part 2: Piecewise linear functions are chosen to be admissible functions only in part of a solution domain; singular (or analytic) functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on their common boundary. We have further developed this method using a new coupling strategy. If L=O(|lnh|), the average errors of numerical solutions and their generalized derivatives are still O(h), where h is the maximal boundary length of regular triangular elements in the finite element method, and L is the total of singular admissible functions in the Ritz-Galerkin method.

The coupling relation L = O(|lnh|) is significant because only a few singular functions are required for a good approximation of solutions.