The nonconforming virtual element method

Abstract

Abstract. We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods.

Some Estimates for Virtual Element Methods

“…The local estimates in Section 2 (and their three-dimensional analogs) are relevant for general second elliptic boundary value problems [8] and nonconforming virtual elements [3]. We also expect that the new techniques can be extended to virtual element methods for higher order problems [15,16].…”

Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems

“…(3.18)- (3.20) in Ref. [4], respectively). They also play a role in determining the high-order part of some post-processings of the potential used in the context of Hybridizable Discontinuous Galerkin methods; cf., e.g., the variation proposed in Ref.…”

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

“…The choice l " k´1 allows one to establish a link (up to equivalent stabilizations) with the high-order MFD method of [6,43] (in the case k " 0, l "´1, one can recover the classical Crouzeix-Raviart element on simplices), while the choice l " k`1 is related to a variant of the HDG method introduced in [42].…”

“…with potential reconstruction operator p k`1 T defined by (6) and stabilization bilinear form j T defined by (10). Introduce now the following global bilinear form obtained by a standard element-by-element assembly procedure:…”

“…This is a distinctive feature with respect to the VE method, where H 1 -conforming reconstructions are present in the background. A study of the relations between HHO and HDG methods can be found in [20], which also fits into the HHO framework (up to equivalent stabilization) the recent high-order MFD method of [6,43] (also referred to as nonconforming VE method in subsequent publications). We also mention that HHO methods for polynomial order k " 0 are closely related to MHFV (and so to lowest-order MFD); cf., in particular, [ HHO methods offer several assets.…”