Improving high-order VEM stability on badly-shaped elements

TL;DR

This paper introduces a new, computationally efficient method to enhance the stability of high-order Virtual Element Methods on poorly-shaped elements by redefining local projectors and degrees of freedom using scaled monomials.

Contribution

It proposes a novel approach that improves VEM stability on badly-shaped elements without high computational costs, using scaled monomials instead of orthonormal bases.

Findings

Improved conditioning of VEM matrices on challenging geometries

Effective in 2D and 3D cases with complex polytopes

Less computationally demanding than previous orthonormal basis methods

Abstract

For the 2D and 3D Virtual Element Methods (VEM), a new approach to improve the conditioning of local and global matrices in the presence of badly-shaped polytopes is proposed. It defines the local projectors and the local degrees of freedom with respect to a set of scaled monomials recomputed on more well-shaped polytopes. This new approach is less computationally demanding than using the orthonormal polynomial basis. The effectiveness of our procedure is tested on different numerical examples characterized by challenging geometries of increasing complexity.