Virtual Element Methods Without Extrinsic Stabilization

TL;DR

This paper introduces new virtual element methods for second order elliptic problems that do not require extrinsic stabilization, achieving optimal error estimates and verified through numerical experiments.

Contribution

It develops VEMs without extrinsic stabilization using $H( extrm{div})$-conforming macro finite element spaces, applicable in arbitrary dimensions.

Findings

Achieved optimal error estimates for the proposed VEMs.

Constructed local $H( extrm{div})$-conforming macro finite element spaces.

Numerical experiments confirm the effectiveness of the methods.

Abstract

Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local $H(\textrm{div})$-conforming macro finite element spaces such that the associated $L^2$ projection of the gradient of virtual element functions is computable, and the $L^2$ projector has a uniform lower bound on the gradient of virtual element function spaces in $L^2$ norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.