Numerical solution of boundary integral equations by means of attenuation factors

We consider first-kind boundary integral equations with logarithmic kernel such as those arising from solving Dirichlet problems for the Laplace equation by means of single-layer potentials. The first-kind equations are transformed into equivalent equations of the second kind which contain the conjugation operator and which are then solved with a degenerate-kernel method based on Fourier analysis and attenuation factors. The approximations we consider, among them spline interpolants, are linear and translation invariant. In view of the particularly small kernel, the linear systems resulting from the discretization can be solved directly by fixed-point iteration.