The role of stabilization in the virtual element method: a survey

TL;DR

This survey reviews the mathematical foundations of stabilization in the virtual element method, highlighting key results, proof techniques, and extensions relevant for newcomers and researchers in the field.

Contribution

It provides a comprehensive overview of stabilization analysis in VEM, including proofs, extensions, and new interpolation estimate methods.

Findings

Summarizes main mathematical results on stabilization in VEM.

Details proofs for 2D nodal conforming and nonconforming VEM.

Introduces a recent interpolation estimate based on stability bounds.

Abstract

The virtual element method was introduced 10 years ago and it has generated a large number of theoretical results and applications ever since. Here, we overview the main mathematical results concerning the stabilization term of the method as an introduction for newcomers in the field. In particular, we summarize the proofs of some results for two dimensional ``nodal'' conforming and nonconforming virtual element spaces to pinpoint the essential tools used in the stability analysis. We discuss their extensions to several other virtual elements. Finally, we show several ways to prove interpolation estimates, including a recent one that is based on employing the stability bounds.