Ill‐conditioning in the virtual element method: Stabilizations and bases

Lorenzo Mascotto

doi:10.1002/num.22257

Abstract

In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.

Highlights

The interest in numerical methods for the approximation of partial differential equations (PDEs in short) based on polytopic grids has grown in the last decade, due to the high flexibility that polygonal/polyhedral meshes allow
We are interested in the behavior of the condition number when varying the stabilization and the polynomial basis dual to internal moments (5) in presence of a sequence of “bad shaped” polygons; this is probed in Sections 3.2 and 3.3
We investigate the behaviour of the condition number of the stiffness matrix associated with method (21) by keeping fixed the meshes of Figure 1 and by increasing p

Summary

Introduction

The interest in numerical methods for the approximation of partial differential equations (PDEs in short) based on polytopic grids has grown in the last decade, due to the high flexibility that polygonal/polyhedral meshes allow. We here recall only a short list including: mimetic finite differences [1, 2], discontinuous Galerkin-finite element method (DG-FEM) [3, 4], hybridizable and hybrid high-order methods [5, 6], weak Galerkin method [7], BEM-based FEM [8], and polygonal FEM [9]. VEM are a generalization of the finite element method (FEM in short) enabling the employment of polytopal meshes and the possibility of building high-order methods. Local VE spaces consist of other functions instrumental for constructing global H1 conforming approximation space; such functions are typically the solution to local PDEs and

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Conclusion