Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains

TL;DR

This paper analyzes the stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains, providing guidelines for choosing charge points to achieve high accuracy and stability, especially at high frequencies.

Contribution

It offers a detailed analysis of charge point placement for the MFS on analytic domains, including nonconvex shapes, and develops a practical recipe for stable, high-accuracy solutions at high frequencies.

Findings

Stable solutions with error norms around 10^{-11} achieved.

High frequency solutions require only about 3 points per wavelength.

Charge point placement depends on domain shape and boundary data.

Abstract

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.