Adaptive finite elements for exterior domain problems

Abstract

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.