Spurious eigenvalue-free algorithms of the method of fundamental solutions for solving the Helmholtz equation in bounded multiply connected domains.

For the Helmholtz equation Δu + k²u = 0 in 2D domain S, there exists a unique solution if k² is not exactly equal to an eigenvalue λ of the Laplace eigenvalue problem Δu + λu = 0 in S. One important criterion for numerical methods is that there must exist no spurious (i.e., superfluous) eigenvalues so that the above unique solution can be obtained correctly. For exterior problems, the method of fundamental solutions (MFS) using Hankel functions suffers from spurious eigenvalues. New modified Hankel functions have been proposed in Zhang et al. (Appl. Numer. Math. 145, 236–260, 2019) to eliminate all spurious eigenvalues. In this paper, we study bounded multiply connected domains. The MFS algorithms without spurious eigenvalues and their strict analysis are our goals. First, we study bounded simply connected domains by the MFS. The algorithms using Hankel functions are free from spurious eigenvalues. A brief error analysis is provided. Next, we focus on bounded multiply connected domains and choose an annular domain for analysis. The Hankel functions and the modified Hankel functions in [47] are chosen as the exterior fundamental solutions (FS) and the interior FS, respectively. Such combined FS eliminate all spurious eigenvalues, and new error and stability analyses are explored. Bounds of errors involve degeneracy, and those of the condition number involve a gap to eigenvalues. Numerical experiments are carried out to support the analysis made, and better pseudo-boundaries of source nodes are also investigated numerically. The new error and stability analyses in this paper are new and essential, thus providing a solid theoretical basis of the MFS for the Helmholtz equation in 2D bounded simply connected and multiply connected domains. [ABSTRACT FROM AUTHOR]