On the sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solve an integral equation of the first kind with logarithmic kernel, the convergence rate is O(h) in the L₂ norm. In this note we show that O(h³) or O(h⁵) convergence in any Sobolev norm (and thus, for example, in L_∞) may be obtained by a simple cheap postprocessing of the original collocation solution. The construction of the postprocessor is based on writing the first kind equation as a second kind equation, and applying the Sloan iteration to the latter equation. The theoretical convergence rates are verified in a numerical example.