Error estimates for numerical solutions of one- and two-dimensional integral equations

This thesis is concerned with the improvement of numerical methods, specifically boundaryelement methods (BEMs), for solving Fredholm integral equations in both one- and two dimensions. The improvements are based on novel (computer-algebra-based) error analyses that yield explicit forms of correction terms for a priori incorporation into BEM methods employing piecewise-polynomial interpolation in the numerical approximation. The work is motivated by
the aim of reducing errors of BEM methods for low-degree interpolating polynomials, without significantly increasing the computational cost associated with higher-degree interpolation. The present thesis develops, implements and assesses improved BEMs on two fronts.

First, a modified Nystr¨om method is developed for the solution of one-dimensional Fredholm integral equations of the second kind (FIE2s). The method is based upon optimal approximation and inclusion of an explicit form of orthogonal-polynomial integration error, and it can be extended to systems of integral equations. It is validated, in both the single and system cases, on challenging
FIE2s that contain a finite number of (integrable) singularities, or points of limited differentiability,
within the integral kernels.

Second, BEMs are developed for solving two-dimensional FIE2s in the widely applicable context of harmonic boundary value problems in which the boundary conditions may be either continuous or discontinuous. In the latter case, modifying the BEM to conquer the adverse effect (on
convergence with decreasing mesh size) caused by boundary singularities requires considerable additional theory and implementation; the motivation for doing so is that such singularities arise naturally in the modelling of, e.g., stress fractures in solid mechanics and dielectrics in
electrostatics. For both non-singular and singular BVPs, standard BEMs are improved herein by optimal approximation and inclusion of explicit forms of Lagrange-interpolation integration errors. The modified BEMs are validated against pseudo-analytic results obtained by a conformal transformation method, for which a novel implementation of the inverse transformation (needed to recover the physical solution) is included explicitly by use of an algebraic manipulator.

Through a set of test problems with known (or otherwise computable) solutions, both the one- and two-dimensional modified methods, for both regular and singular BVPs, are demonstrated to show marked improvements in performance over their unmodified counterparts.