Stabilization of the nonconforming virtual element method

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.

Automated summary

In this paper, we address the issue of designing robust stabilization terms for the nonconforming virtual element method. We transfer the problem of defining the stabilizing bilinear form to the dual space, which allows us to construct different bilinear forms with optimal or quasi-optimal stability bounds and error estimates. This approach relaxes the assumptions on the tessellation and proves optimality even under geometrical conditions that allow a mesh to have a very large number of arbitrarily small edges per element. We also numerically assess the performance of the VEM under different stabilizations using representative test cases.