Anisotropic Error Estimates of The Linear Nonconforming Virtual Element Methods

TL;DR

This paper develops a refined error analysis for the linear nonconforming Virtual Element Method (VEM) applied to Poisson problems, introducing new geometric assumptions and stabilization techniques for anisotropic meshes in 2D and 3D.

Contribution

It introduces new geometric assumptions and a novel stabilization approach for anisotropic elements in the nonconforming VEM, enhancing error analysis accuracy.

Findings

New error equation for linear nonconforming VEM derived

Stabilization using projection on extended element patch introduced

Error estimates refined for anisotropic polytopal meshes

Abstract

A refined a priori error analysis of the lowest order (linear) nonconforming Virtual Element Method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on shape regularity of polytopal meshes. A new error equation for the lowest order (linear) nonconforming VEM is derived for any choice of stabilization, and a new stabilization using a projection on an extended element patch is introduced for the error analysis on anisotropic elements.