A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations

Abstract 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. (2019) 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.