Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc, which is a reformulation of the Dirichlet problem of the Laplace equation in the plane. The optimal convergence order and quasi-linear complexity order of the proposed method are established. A precondition is introduced. Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc, we obtain the solution of the Dirichlet problem on a smooth open arc in the plane. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.