Quantitative study of the stabilization parameter in the virtual element method

TL;DR

This paper investigates the role of the stabilization parameter in the virtual element method (VEM), linking it to generalized barycentric coordinates and isoparametric finite elements to better understand its influence on solution stability and accuracy.

Contribution

It establishes a new connection between VEM stabilization and barycentric coordinates, offering a novel interpretation of stability as energy of specific functions in the discrete space.

Findings

Stability linked to energy of functions in discrete space

Provides insights into stabilization's impact on solution behavior

Connects VEM stabilization with finite element concepts

Abstract

The choice of stabilization term is a critical component of the virtual element method (VEM). However, the theory of VEM provides only asymptotic guidance for selecting the stabilization term, which ensures convergence as the mesh size approaches zero, but does not provide a unique prescription for its exact form. Thus, the selection of a suitable stabilization term is often guided by numerical experimentation and analysis of the resulting solution, including factors such as stability, accuracy, and efficiency. In this paper, we establish a new link between VEM and generalized barycentric coordinates, in particular isoparametric finite elements as a specific case. This connection enables the interpretation of the stability as the energy of a particular function in the discrete space, commonly known as the `hourglass mode.' Through this approach, this study sheds light on how the virtual element solution depends on the stabilization term, providing insights into the behavior of the method in more general scenarios.