A combination of spectral and finite elements for an exterior problem in the plane

Abstract

We present two Galerkin BEM–FEM schemes based on a combination of spectral and finite element methods to solve numerically an exterior Poisson problem in the plane. We provide error estimates for the Galerkin methods, propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration still preserves the rate of convergence. The advantage of using a spectral approximation for the unknown defined on the boundary is that few degrees of freedom are needed on this interface. Therefore, the global systems of linear equations associated to the fully discrete problems are easier to solve as we show in our numerical experiments.