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.

翻译：本文基于[5]中提出的全局与局部通量重构思想，为虚拟元方法（VEM）提出了两种后验误差估计器。所提出的误差估计器对VEM的$h$版本、$p$版本以及$hp$版本均具有可靠性与高效性。这解决了我们先前在[6]中基于求解局部非混合化混合问题的方法存在的部分局限性。与有限元情形不同，效率性的证明过程更为简洁，因为VEM中的通量重构不要求存在多项式、稳定、散度右逆算子。相反，我们仅需利用虚拟元函数的隐式定义，在虚拟元空间中构造右逆算子。理论结果通过一个基准问题上的若干数值实验得到了验证。