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：我们提出了基于[5] 精神的全球和本地通量重建的虚拟元件方法(VEM)的事后误差估计值。提议的误差估计值对于VEM的美元-、美元-和美元-和美元-版本是可靠和高效的。这解决了我们以前在[6]中的方法的局部限制,而[6]中的方法基于解决当地非湿化混合问题。与有限要素设置不同,效率的证明证明更简单,因为VEM的通量重建并不要求存在多元、稳定、分立右反操作器。相反,我们只需要在虚拟元件空间建造右反操作器,只利用虚拟元函数的隐含定义。理论结果通过一些关于基准问题的数字实验得到验证。