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

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Abstract
The aim of this paper is to develop a straightforward analysis of the Galerkin method for two-dimensional boundary integral equations of the first kind with logarithmic kernels. A distinctive feature of the analysis is that no appeal is made to ‘coercivity’, as a result of which some existence questions cannot be answered directly. In return, however, the analysis has no special difficulty in handling corners, cusps, or open arcs. Instead of coercivity, the central feature of the analysis is the positive-definite property 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 companion paper Sloan & Spence (1987) for problems on both open and closed polygonal arcs.

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