Exploring High-order three dimensional Virtual Elements: bases and stabilizations
For researchers using VEM, this work addresses numerical stability issues in high-order 3D simulations, but the improvements are incremental.
The paper tests high-order Virtual Element Methods (up to p=10) for 3D Poisson problems, identifying causes of suboptimal convergence (e.g., ill-conditioning, poor stabilization) and proposing improved degree-of-freedom definitions and stabilizations that enhance robustness.
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.