Superconvergent gradient recovery for virtual element methods

Abstract

Virtual element method is a new promising finite element method using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of virtual element methods by doing some local post-processing only on the degrees of freedom. Using the linear virtual element method as an example, we propose a universal gradient recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a posteriori error estimators in adaptive computation is also investigated. Compared to the existing residual-type a posteriori error estimators for the virtual element methods, the recovery-type a posteriori error estimator based on the proposed gradient recovery technique is much simpler in implementation and it is asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator.