Stability Analysis for the Virtual Element Method

TL;DR

This paper extends the analysis of Virtual Element Methods (VEM) to more general meshes, including those with very small edges, and introduces a simplified stabilization approach that maintains stability and convergence.

Contribution

The paper provides a stability analysis of VEM on general meshes, introduces a simplified stabilization form, and validates the approach with numerical tests.

Findings

VEM can handle meshes with arbitrarily small edges.

A simplified stabilization involving only boundary degrees of freedom is effective.

Numerical results confirm theoretical stability and convergence.

Abstract

We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.