A discrete collocation method for Symmʹs integral equation on curves with corners

This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symmʹs equation) on a closed polygonal boundary in ℝ2. We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.