Error Estimates for the Finite Element Approximation of Problems in Unbounded Domains

In this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi-infinite strip in the plane. The finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition. In fact there is a family of approximate boundary conditions with increasing accuracy (and computational cost) for a given artificial boundary. Our error estimates are based on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. Numerical examples for Poisson’s problem outside a circle and in a semi-infinite strip are presented. Numerical results demonstrate the performance of our error estimates.