Adaptive virtual element methods with equilibrated fluxes

TL;DR

This paper develops an hp-adaptive virtual element method for diffusion problems, introducing a reliable a posteriori error estimator and flux reconstruction techniques, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a new hp-adaptive VEM with an efficient a posteriori error estimator and localized flux reconstruction, ensuring p-robustness and satisfying a discrete inf-sup condition.

Findings

The mixed VEM satisfies a discrete inf-sup condition independent of discretization.

The proposed error estimator is reliable, efficient, and p-robust.

Numerical experiments confirm the theoretical properties and compare with residual estimators.

Abstract

We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition, with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM, with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.