VEM and the Mesh

TL;DR

This paper investigates how the shape of polygonal elements in meshes affects the performance of the virtual element method (VEM), including experimental convergence analysis and the development of a mesh quality indicator.

Contribution

It provides new insights into the impact of mesh shape irregularities on VEM performance and introduces a mesh quality metric to predict method effectiveness.

Findings

VEM convergence is affected by shape irregularities.

A correlation exists between mesh quality metrics and VEM performance.

A new mesh quality indicator can predict VEM performance on given meshes.

Abstract

In this work we report some results, obtained within the framework of the ERC Project CHANGE, on the impact on the performance of the virtual element method of the shape of the polygonal elements of the underlying mesh. More in detail, after reviewing the state of the art, we present a) an experimental analysis of the convergence of the VEM under condition violating the standard shape regularity assumptions, b) an analysis of the correlation between some mesh quality metrics and a set of different performance indexes, and c) a suitably designed mesh quality indicator, aimed at predicting the quality of the performance of the VEM on a given mesh.