Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. (English) Zbl 1170.65082

Barnett, A. H.; Betcke, T.

J. Comput. Phys. 227, No. 14, 7003-7026 (2008).

This paper is concerned with the numerical approximation of solutions to the Helmholtz equation \(\Delta u+k^2 u=0\) in \(\Omega\), subject to Dirichlet boundary condition \(u=v\) on \(\partial\Omega\). Here \(\Omega\subset {\mathbb C}\) is a simply connected domain with analytic boundary. The idea of the proof is to use the Method of Fundamental Solutions (MFS), that is, to approximate the solution \(u\) as a linear combination of fundamental solutions, \(\sum_{j=1}^N \alpha_j H^{(1)}_0(k|x-y_j|)\) with \(y_j\in{\mathbb C}\setminus\overline\Omega\), where \(H^{(1)}_0\) is the Hankel function of the first kind with order zero.
The main goal of the paper is to investigate conditions on the charges \(y_j\) that lead to an accurate and stable numerical scheme in the framework of the MFS. The authors emphasize that this fact depends on how far into the complex plane the solution \(u\) can be analytically continued. More precisely, it is obtained in the paper that for a general analytic domain \(\Omega\), the position of the charge points relative to the singularities of the analytic continuation of \(u\) is crucial for the accuracy and numerical stability of the MFS. Various numerical experiments on many domain shapes are presented that support the conclusion.

Reviewer: Marius Ghergu (Dublin)

MSC:

65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78M25	Numerical methods in optics (MSC2010)
78A25	Electromagnetic theory, general

Keywords:

Helmholtz equation; method of fundamental solutions; analytic continuation; charge curve; stability; convergence; high frequency waves; numerical experiments

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