  Since its introduction, the Virtual Element Method (VEM) was shown to be able
to deal with a large variety of polygons, while achieving good convergence
rates. The regularity assumptions proposed in the VEM literature to guarantee
the convergence on a theoretical basis are therefore quite general. They have
been deduced in analogy to the similar conditions developed in the Finite
Element Methods (FEMs) analysis. In this work, we experimentally show that the
VEM still converges with almost optimal rates and low errors in the L2 and H1
norms even if we significantly break the regularity assumptions that are used
in the literature. These results suggest that the regularity assumptions
proposed so far might be overestimated. We also exhibit examples on which the
VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality
indicator that experimentally correlates the entity of the violation of the
regularity assumptions and the performance of the VEM solution, thus predicting
if a dataset is potentially critical for VEM.
