Benchmarking the geometrical robustness of a Virtual Element Poisson solver

Polytopal Element Methods (PEM) allow us solving differential equations on generalpolygonal and polyhedral grids, potentially offering great flexibility to meshgeneration algorithms. Differently from classical finite element methods, wherethe relation between the geometric properties of the mesh and the performances ofthe solver are well known, the characterization of a good polytopal element is stillsubject to ongoing research. Current shape regularity criteria are quite restrictive,and greatly limit the set of valid meshes. Nevertheless, numerical experiments revealedthat PEM solvers can perform well on meshes that are far outside the strictboundaries imposed by the current theory, suggesting that the real capabilities ofthese methods are much higher. In this work, we propose a benchmark to studythe correlation between general 2D polygonal meshes and PEM solvers which wetest on a virtual element solver for the Poisson equation. The benchmark aims toexplore the space of 2D polygonal meshes and polygonal quality metrics, in orderto understand if and how shape regularity, defined according to different criteria,affects the performance of the method. The proposed tool is quite general, andcan be potentially used to study any PEM solver. Besides discussing the basics ofthe benchmark, we demonstrate its application on a representative member of thePEM family, namely the Virtual Element Method, also discussing our findings.