Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2- and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented.