Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions

TL;DR

This paper introduces reliable and efficient a posteriori error estimators for the virtual element method, improving flux reconstruction techniques to enhance adaptive algorithms across various VEM versions.

Contribution

It develops new flux-based error estimators for VEM that are simpler to prove and more broadly applicable than previous methods, addressing prior limitations.

Findings

Estimators are reliable and efficient for h-, p-, and hp-versions of VEM.

Flux reconstruction does not require polynomial divergence right-inverse operators.

Numerical experiments validate the theoretical results.

Abstract

We present two a posteriori error estimators for the virtual element method (VEM) based on global and local flux reconstruction in the spirit of [5]. The proposed error estimators are reliable and efficient for the $h$-, $p$-, and $hp$-versions of the VEM. This solves a partial limitation of our former approach in [6], which was based on solving local nonhybridized mixed problems. Differently from the finite element setting, the proof of the efficiency turns out to be simpler, as the flux reconstruction in the VEM does not require the existence of polynomial, stable, divergence right-inverse operators. Rather, we only need to construct right-inverse operators in virtual element spaces, exploiting only the implicit definition of virtual element functions. The theoretical results are validated by some numerical experiments on a benchmark problem.