Stabilization of the nonconforming virtual element method

TL;DR

This paper develops a new framework for designing stabilization terms in the nonconforming virtual element method, enabling optimal stability and error estimates even on highly irregular meshes.

Contribution

It introduces a dual space approach for constructing stabilization bilinear forms, allowing for weaker mesh assumptions and improved robustness.

Findings

Achieved optimal stability bounds under relaxed mesh assumptions

Constructed various stabilizations with optimal or quasi-optimal error estimates

Numerical tests confirm the effectiveness of the new stabilization framework

Abstract

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.