Superconvergent Gradient Recovery for Virtual Element Methods

TL;DR

This paper introduces a universal gradient recovery technique for virtual element methods that enhances gradient accuracy, demonstrates superconvergence, and provides a simpler, asymptotically exact a posteriori error estimator for adaptive mesh refinement.

Contribution

It proposes a new gradient recovery procedure for virtual element methods that improves accuracy and simplifies a posteriori error estimation.

Findings

Demonstrates superconvergence of the recovered gradient.

Shows the recovery-based error estimator is simpler and asymptotically exact.

Validates the method through benchmark tests.

Abstract

Virtual element methods is a new promising finite element methods 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 the virtual element methods by doing some local post-processing only on the degrees of freedom. Using linear virtual element method as an example, we propose a universal 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 methods 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 asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotical exactness of the recovery-based a posteriori error estimator.