Exploring High-order three dimensional Virtual Elements: bases and stabilizations

TL;DR

This paper investigates high-order 3D Virtual Element Methods for Poisson problems, identifies issues affecting convergence, and proposes improved stabilization techniques to enhance numerical robustness.

Contribution

It introduces new stabilization variants and face/bulk degree definitions that improve the robustness and accuracy of high-order 3D Virtual Element Methods.

Findings

Enhanced stability with proposed stabilization variants

Improved convergence behavior in numerical tests

Reduced ill-conditioning of the stiffness matrix

Abstract

We present numerical tests of the Virtual Element Method (VEM) tailored for the discretization of a three dimensional Poisson problem with high-order "polynomial" degree (up to $p=10$). Besides, we discuss possible reasons for which the method could return suboptimal-wrong error convergence curves. Among these motivations, we highlight ill-conditioning of the stiffness matrix and not particularly "clever" choices of the stabilizations. We propose variants of the definition of face/bulk degrees of freedom, as well as of stabilizations, which lead to methods that are much more robust in terms of numerical performances.