Exploring high-order three dimensional virtual elements: Bases and stabilizations

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.