The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.