Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions
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 -, -, and -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.
Cite
@article{arxiv.2107.03716,
title = {Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions},
author = {F. Dassi and J. Gedicke and L. Mascotto},
journal= {arXiv preprint arXiv:2107.03716},
year = {2025}
}
Comments
In the proof of Theorem 4.3 there is a mistake. More precisely, when we use the problem (27), we want to exploit the fact that $(div(\sigma), v) = (r^k,v)$ for all functions $v$ including the constant function. This is however true only if $r^k$ has zero integral mean, which we forgot to check and is in fact not true in general