An optimal order collocation method for first kind boundary integral equations on polygons

Abstract

This paper discusses the convergence of the collocation method using splines of any order $k$ for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ${\Bbb R}^2$ . Before discretization the equation is transformed to an equivalent equation over $[-\pi,\pi]$ using a nonlinear parametrization of the polygon which varies more slowly than arc--length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on $[-\pi,\pi]$ . This latter integral equation is shown to be well--posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non--standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order $k$ (degree $k-1$ ) converges with optimal order $O (h^k)$ . (The collocation points are the mid--points of subintervals when $k$ is odd and the break--points when $k$ is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples.