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

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.