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

The virtual element method was introduced 10 years ago, and it has generated ever since a large number of theoretical results and applications. Here, we give an overview of the main mathematical results concerning the stabilization term of the method in the hope that it may be useful to newcomers in the field. In particular, we summarize the proof 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 extension to several other virtual elements. Finally, we show that the stability bounds imply interpolation estimates.