BOUNDARY INTEGRAL-EQUATIONS IN TIME-HARMONIC ACOUSTIC SCATTERING

We first review the basic existence results for exterior boundary value problems for the Helmholtz equation via boundary integral equations. Then we describe the numerical solution of these integral equations in two dimensions for a smooth boundary curve using trigonometric polynomials on an equidistant mesh. We provide a comparison of the Nystrom method, the collocation method and the Galerkin method. In each case we take proper care of the logarithmic singularity of the kernel of the integral equation by choosing appropriate quadrature rules. In the case of analytic data the convergence order is exponential. The Nystrom method is the most efficient since it requires the least computational effort. Finally, we consider boundary curves with corners. Here, we use a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an appropriate substitution. Then, the application of Nystrom's method again yields rapid convergence.