Polyhedral Mesh Quality Indicator for the Virtual Element Method

TL;DR

This paper introduces a mesh quality indicator that predicts the Virtual Element Method's performance on complex polyhedral meshes, aiding in understanding convergence behavior and mesh suitability.

Contribution

It presents a novel mesh quality indicator that measures regularity violations and correlates with VEM performance on diverse polyhedral meshes.

Findings

Strong correlation between the indicator and VEM accuracy

Effective prediction of VEM behavior on non-convex meshes

Validation on a wide range of mesh geometries

Abstract

We present the design of a mesh quality indicator that can predict the behavior of the Virtual Element Method (VEM) on a given mesh family or finite sequence of polyhedral meshes (dataset). The mesh quality indicator is designed to measure the violation of the mesh regularity assumptions that are normally considered in the convergence analysis. We investigate the behavior of this new mathematical tool on the lowest-order conforming approximation of the three-dimensional Poisson equation. This work also assesses the convergence rate of the VEM when applied to very general polyhedral meshes, including non-convex and skewed three-dimensional elements. Such meshes are created within an original mesh generation framework, which is designed to allow the generation of meshes with very different sizes, connectivity and geometrical properties. The obtained results show a significant correlation between the quality measured a priori by the indicator and the effective performance of the VEM.