High-order Virtual Element Method on polyhedral meshes

TL;DR

This paper evaluates the high-order Virtual Element Method for 3D diffusion-reaction problems, focusing on convergence, mesh types, stability, and order sensitivity, providing insights into its numerical performance on complex polyhedral meshes.

Contribution

It introduces a comprehensive numerical assessment of high-order VEM in three dimensions, highlighting its convergence properties and robustness on irregular polyhedral meshes.

Findings

Demonstrates h-convergence for various polynomial orders

Shows robustness on irregular polyhedral Voronoi meshes

Analyzes sensitivity to stabilization parameters

Abstract

We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher "polynomial" order k and three space dimensions. Although the main focus of the present study is to illustrate some h-convergence tests for different orders k, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order k.