  Polytopal Element Methods (PEM) allow to solve differential equations on
general polygonal and polyhedral grids, potentially offering great flexibility
to mesh generation algorithms. Differently from classical finite element
methods, where the relation between the geometric properties of the mesh and
the performances of the solver are well known, the characterization of a good
polytopal element is still subject to ongoing research. Current shape
regularity criteria are quite restrictive, and greatly limit the set of valid
meshes. Nevertheless, numerical experiments revealed that PEM solvers can
perform well on meshes that are far outside the strict boundaries imposed by
the current theory, suggesting that the real capabilities of these methods are
much higher. In this work, we propose a benchmark to study the correlation
between general 2D polygonal meshes and PEM solvers. The benchmark aims to
explore the space of 2D polygonal meshes and polygonal quality metrics, in
order to identify weaker shape-regularity criteria under which the considered
methods can reliably work. The proposed tool is quite general, and can be
potentially used to study any PEM solver. Besides discussing the basics of the
benchmark, in the second part of the paper we demonstrate its application on a
representative member of the PEM family, namely the Virtual Element Method,
also discussing our findings.
