Spectral Approximation of Time-harmonic Maxwell Equations in Three-dimensional Exterior Domains

We develop in this paper an efficient and robust spectral-Galerkin method to solve the three-dimensional time-harmonic Maxwell equations in exterior domains. We first reduce the problem to a bounded domain by using the capacity operator which characterizes the nonreflecting boundary condition. Then, we adapt the transformed field expansion (TFE) approach to reduce the problem to a sequence of Maxwell equations in a separable spherical shell. Finally, we develop an efficient spectral algorithm by using Legendre approximation in the radial direction and vector spherical harmonic expansion in the tangential directions.