Ill-conditioning in the Virtual Element Method: stabilizations and bases
For researchers using the Virtual Element Method, this work addresses a practical numerical stability issue, but the findings are incremental and limited to 2D problems.
The paper investigates ill-conditioning in the stiffness matrix for 2D Poisson problems solved with the Virtual Element Method, showing that high-order methods and bad-shaped polygons cause ill-conditioning. It demonstrates that modifying internal moments with non-standard polynomials improves the condition number, while standard stabilizations yield similar results.
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.