The role of mesh quality and mesh quality indicators in the Virtual Element Method

TL;DR

This paper demonstrates that the Virtual Element Method (VEM) converges well even when regularity assumptions are significantly violated, and introduces a mesh quality indicator to predict potential issues.

Contribution

The study shows VEM's robustness beyond traditional regularity assumptions and proposes a new mesh quality indicator for assessing dataset suitability.

Findings

VEM converges with near-optimal rates despite breaking regularity assumptions.

A mesh quality indicator correlates with VEM performance and potential convergence issues.

Examples of sub-optimal convergence and divergence are provided.

Abstract

Since its introduction, the Virtual Element Method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the Finite Element Methods (FEMs) analysis. In this work, we experimentally show that the VEM still converges with almost optimal rates and low errors in the L2 and H1 norms even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a dataset is potentially critical for VEM.