High-order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs

This paper is devoted to the solution of the Dirichlet problem for the Laplace and Helmholtz equation in the complement of a smooth open curve in the plane. The solution is looked for as a single-layer potential, the corresponding density being therefore the solution of an integral equation on the open arc. This equation is transformed into an equivalent 1-periodic integral equation having existence and uniqueness of solution for any periodic data. Here we study the use of collocation and quadrature methods for solving this equation. We show the convergence of both methods and prove the existence of an asymptotic expansion of the error in powers of the discretization parameter. As a consequence we show that for some of them a superconvergence phenomenon occurs when computing the solution of the differential problem. Two numerical experiments are shown in order to illustrate the theoretical results introduced in this work.