Recovery-based error estimators for the VEM and BEM-based FEM

Abstract

In this article, we consider recovery-based a posteriori error estimators for the Virtual Element Method (VEM) and the Boundary Element Method based Finite Element Method (BEM-based FEM). Both methods are Galerkin methods on polygonal and polyhedral grids for the numerical solution of partial differential equations. Thus, they are highly flexible and particularly efficient in combination with adaptive refinement. Our error estimator computes the distance between the finite element gradient and its post-processed version to obtain an accurate approximation of the true error. The post-processing is realised by local averaging and is therefore easy to implement and fast. Furthermore, we have found points of extraordinary accuracy, so called stress points, on specific regular elements for the BEM-based FEM. We demonstrate that, when sampling these stress points, the recovered gradient becomes superconvergent, which means that it converges at a greater rate than the unprocessed gradient.