A posteriori error estimates for the finite element discretization of second-order PDEs set in unbounded domains

Abstract

We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion equation as well as Helmholtz problems in waveguides with perfectly matched layers. The usual procedure to deal with such problems is to first consider a modeling error due to the introduction of the artificial boundary, and estimate the remaining discretization error with a standard a posteriori technique. A shortcoming of this method, however, is that it is typically hard to obtain sharp bounds on the modeling error. In this work, we propose a new technique that allows to control the whole error by an a posteriori error estimator. Specifically, we propose a flux-equilibrated estimator that is slightly modified to handle the truncation boundary. For the reaction diffusion equation, we obtain fully-computable guaranteed error bounds, and the estimator is locally efficient and polynomial-degree-robust provided that the elements touching the truncation boundary are not too refined. This last condition may be seen as an extension of the notion of shape-regularity of the mesh, and does not prevent the design of efficient adaptive algorithms. For the Helmholtz problem, as usual, these statements remain valid if the mesh is sufficiently refined. Our theoretical findings are completed with numerical examples which indicate that the estimator is suited to drive optimal adaptive mesh refinements.