Optimal convergence rates of an adaptive finite element method for unbounded domains

We consider linear reaction-diffusion equations posed on unbounded domains, and discretized by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, it is necessary to introduce a truncation boundary, whereby only a bounded computational subdomain is meshed, leading to an approximation of the solution by zero in the remainder of the domain. We propose a residual-based error estimator that accounts for both the standard discretization error as well as the effect of the truncation boundary. This estimator is shown to be reliable and efficient under appropriate assumptions on the triangulation. Based on this estimator, we devise an adaptive algorithm that automatically refines the mesh and pushes the truncation boundary towards infinity. We prove that this algorithm converges and even achieves optimal rates in terms of the number of degrees of freedom. We finally provide numerical examples illustrating our key theoretical findings.