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

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 exterior solution 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 efficient evaluation of the discrete DtN map by the fast Fourier transform (FFT). By incorporating the boundary condition defined by the DtN map into a finite element formulation, the original exterior problem posed on an unbounded domain is reduced to an equivalent problem on a bounded computational domain, which can then be solved by the finite element method (FEM). Numerical examples show that the proposed MFS-based approach computes the DtN map with high accuracy, and that the FEM incorporating the boundary condition defined by the DtN map yields accurate solutions for exterior Helmholtz problems with an inhomogeneous term posed on unbounded domains.