Adaptive finite elements for exterior domain problems

Summary.

We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in ${\Bbb R}^d$, $d\in\{2,3\}$. We describe a procedure to generate a sequence of bounded computational domains $\Omega_h^k$, $k=1,2,...$, more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution $u_h$ is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence.