FFT-accelerated computation of the Dirichlet-to-Neumann map for inhomogeneous exterior Helmholtz problems using the method of fundamental solutions

Abstract

An efficient numerical method is proposed for computing the Dirichlet-to-Neumann (DtN) map associated with the exterior Dirichlet problem for the two-dimensional Helmholtz equation with an inhomogeneous term. The solution in the subdomain exterior to a circular artificial boundary is approximated by the method of fundamental solutions (MFS). When the source and collocation points are equally spaced on concentric circles, the coefficient matrices arising in the discretization become circulant, which enables the efficient evaluation of the discrete DtN map by the fast Fourier transform (FFT). By incorporating the transparent boundary condition (TBC) defined by the DtN map into a finite element formulation, the original exterior problem posed in an unbounded domain is reduced to an equivalent problem in a bounded computational domain, which can then be solved by the finite element method (FEM). Numerical examples show that the proposed MFS-based approach provides a highly accurate approximation of the DtN map, that the FFT-based algorithm enables the efficient computation of the resulting discrete DtN map, and that the FEM incorporating the TBC defined by the DtN map yields accurate solutions for exterior Helmholtz problems posed in unbounded domains with an inhomogeneous term.

arXiv comment36 pages, 21 figures. Accepted for publication in Mathematics and Computers in Simulation