Conforming and nonconforming virtual element methods for elliptic problems

TL;DR

This paper introduces a unified framework for conforming and nonconforming Virtual Element Methods (VEM) to solve second order elliptic problems, providing stability conditions and confirming optimal error estimates through numerical experiments.

Contribution

It develops a unified approach for conforming and nonconforming VEM for elliptic problems, establishing stability conditions and optimal error estimates.

Findings

Optimal $H^1$- and $L^2$-error estimates achieved

Numerical experiments confirm comparable accuracy of both methods

Framework applicable to 2D and 3D polygonal meshes

Abstract

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal $H^1$- and $L^2$-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.