Richardson Extrapolation Applied to the Numerical Solution of Boundary Integral Equations for a Laplace’s Equation Dirichlet Problem

Richardson extrapolation is applied to improve the accuracy of the numerical solution of boundary integral equations. The boundary integral equations arise from a direct boundary integral method for solving a Laplace’s equation interior Dirichlet problem. Specifically, the Richardson extrapolation is used to improve the accuracy of collocation. Numerical justification is provided to support the expectation of improved accuracy. The order of the dominant collocation error term is numerically estimated and numerical results are obtained for a simple model problem. Introduction and statement of problem The problem of interest involves the numerical solution of an interior Dirichlet problem for Laplace’s equation on a rectangular domain. The method of interest is a direct boundary integral method. The boundary integral equations are discretized using collocation, and numerically solved for the unknown outward normal boundary flux (normal derivative of the primary unknown). The discretized boundary integral equations, which are Fredholm integral equations of the first kind, are the boundary element equations. In particular, the emphasis is on efficiently improving the numerical solution of the boundary element equations. That is, we seek a more accurate numerical approximation for the outward normal boundary flux. Then, a more accurate numerical solution for the primary unknown in the domain interior can be computed using this more accurate result for the outward normal boundary flux. There are papers presenting collocation convergence and error estimation for Fredholm integral equations of the first kind, for example 2, . However, new material that we present in this paper is an enhancement of pointwise approximations along the boundary of the domain. Richardson extrapolation is applied to improve the accuracy of the numerical solution of the boundary integral equations. Specifically, Richardson extrapolation is used to improve the accuracy of the collocation results. The use of a numerical approximation of the rate of convergence, i.e., the order of the dominant error term of the normal boundary flux approximation, (as the boundary grid is refined) in order to justify application of Richardson extrapolation and the manner in which Richardson extrapolation is applied are contributions of this paper. A Mathematica notebook implementing the boundary element method was written by the author.