This paper is concerned with developing a medius error analysis for several nonconforming virtual element methods (VEMs) for the Poisson equation and the biharmonic equation in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). After some technical derivation, the inverse inequalities and norm equivalence are derived for some conforming VEMs. With the help of these results and following some ideas in Gudi (2010), we obtain medius error estimates for the nonconforming VEMs under discussion, which are optimal up to the regularity of the weak solution. Such estimates also imply that the nonconforming VEMs are convergent even if the exact solution only belongs to the admissible space while the right-hand side of the related equation has some additional regularity.

本文涉及为二维泊松方程和双调和方程的几种不合格虚拟元素方法（VEM）开发medius误差分析，其多边形网格族满足Brezzi等人给出的非常通用的几何假设。（2009）和Chen and Huang（2018）。经过一些技术推导，得出了一些符合的VEM的逆不等式和范数等价。借助这些结果并遵循Gudi（2010）中的一些想法，我们获得了所讨论的不合格VEM的medius误差估计，在弱解的正则性条件下是最优的。