Extrapolation techniques and the collocation method for a class of boundary integral equations

In this paper we prove the existence of asymptotic expansions of the error of the spline collocation method applied to Fredholm integral equations of the first kind with logarithmic kernels. These expansions justify the use of Richardson extrapolation for the acceleration of convergence of the method. The results are stated and proven for a single equation, corresponding to the parameterization of a boundary integral equation on a smooth closed curve. As a byproduct we obtain the nodal superconvergence of the scheme. These results are then extended to smooth open arcs and to systems of integral equations. Finally we prove that such expansions also exist for the Sloan iteration of the numerical solution. 1