Retarded Boundary Integral Equations on the Sphere: Exact and Numerical Solution

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation (RPIE) on the bounded surface of the scatterer. We formulate an algorithm for the spacetime Galerkin discretization with smooth and compactly supported temporal basis functions which have been introduced in [S. Sauter and A. Veit: A Galerkin Method for Retarded Boundary Integral Equations with Smooth and Compactly Supported Temporal Basis Functions, Preprint 04-2011, Universität Zürich]. For the debugging of an implementation and for systematic parameter tests it is essential to have some explicit representations and some analytic properties of the exact solutions for some special cases at hand. We will derive such explicit representations for the case that the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed method. AMS subject classifications. 35L05, 65R20