A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains

We extend the fictitious domain spectral method presented in Gu and Shen (SIAM J Sci Comput 43:A309–A329, 2021) for elliptic PDEs in bounded domains to the Helmhotlz equation in exterior domains. We first reduce the problem in an exterior domain to a bounded domain using the exact Dirichlet-to-Neumann operator. Next, we formulate the reduced problem into an equivalent problem in an annulus by using a fictitious domain approach. Then, we apply the Fourier-spectral method in the radial direction to reduce the problem in an annulus to a sequence of 1-D Bessel-type equations, each with a one-sided open boundary condition that are to be determined by the boundary condition of the original Helmholtz equation. We solve these 1-D Bessel-type equations by the Legendre-spectral method, and determine the open boundary conditions with a least square approach. We derive a wave number explicit error estimate for the special case of a circular obstacle, and provide ample numerical results to show the effectiveness of the proposed method.