T. Betcke

A. H. Barnett, T. Betcke

The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10^{-11} on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods.

T. Betcke, N. Salles, W. Śmigaj

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge-Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows to overresolve in the Laplace domain by computing more Laplace domain solutions solutions than there are time steps. We use techniques from complex approximation theory to analyse the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results.