2023/04/21 by L. Mascotto, Mascotto, L.
Engineering · Mathematics · #65N12 #65N15 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2304.10968
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.