Asymptotic stability of the fundamental solution method
ISCM '90 Proceedings of the International Symposium on Computation mathematics
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Optimization of spectral functions of Dirichlet-Laplacian eigenvalues
Journal of Computational Physics
SIAM Journal on Scientific Computing
The method of fundamental solutions for Helmholtz-type equations in composite materials
Computers & Mathematics with Applications
Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
SIAM Journal on Numerical Analysis
A multilayer method of fundamental solutions for Stokes flow problems
Journal of Computational Physics
Boundary particle method for Laplace transformed time fractional diffusion equations
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
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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^-^1^1 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.