Computing Acoustic Waves in an Inhomogeneous Medium of the Plane by a Coupling of Spectral and Finite Elements

In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition, which consists in a linear integral equation. This integral equation is discretized by a spectral Galerkin approximation method. We provide error estimates for the Galerkin method, propose fully discrete schemes based on elementary quadrature formulas, and show that the perturbation due to this numerical integration gives rise to a quasi-optimal rate of convergence. We also suggest a method for implementing the algorithm using the preconditioned GMRES method and provide some numerical results.