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2007.107: Stability and Convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains

2007.107: A. H. Barnett and T. Betcke (2008) Stability and Convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains. Journal of Computational Physics, 227 (14). pp. 7003-7026. ISSN 0021-9991

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DOI: 10.1016/j.jcp.2008.04.008

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.

Item Type:Article
Uncontrolled Keywords:Helmholtz equation, boundary value problem, Method of Fundamental Solutions, analytic continuation, high frequency waves
Subjects:MSC 2000 > 65 Numerical analysis
MSC 2000 > 78 Optics, electromagnetic theory
MIMS number:2007.107
Deposited By:Dr. Timo Betcke
Deposited On:17 June 2008

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