Lowest order stabilization free Virtual Element Method for the 2D Poisson equation

TL;DR

This paper presents a novel first-order virtual element method for the 2D Poisson equation that eliminates the need for stabilization terms by leveraging higher order polynomial projections, with proven stability and convergence.

Contribution

It introduces the E^2VEM, a stabilization-free virtual element method utilizing enlarged enhancement properties for improved efficiency and theoretical guarantees.

Findings

Method achieves optimal convergence rates.

Numerical tests confirm theoretical stability and accuracy.

Applicable to convex and non-convex polygonal meshes.

Abstract

We introduce and analyse the first order Enlarged Enhancement Virtual Element Method (E$^2$VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement (from which comes the prefix of the name E$^2$) property of local virtual spaces. The polynomial degree of local projections is chosen based on the number of vertices of each polygon. We provide a proof of well-posedness and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the criterium for well-posedness and the theoretical convergence rates.