# Ill-conditioning in the Virtual Element Method: stabilizations and bases

## TL;DR

This paper analyzes the ill-conditioning of the stiffness matrix in the 2D Virtual Element Method for Poisson problems, showing how basis modifications and stabilization choices affect numerical stability.

## Contribution

It demonstrates how to improve the condition number by modifying internal moments and compares stabilization strategies for better numerical stability.

## Key findings

- Ill-conditioning worsens with high-order methods and irregular polygons.
- Modifying internal moments can improve the condition number.
- Standard stabilization choices yield similar condition numbers.

## Abstract

In this paper 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.