Refinement strategies for polygonal meshes applied to adaptive VEM discretization

In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well and states new issues, here tackled, concerning good quality mesh elements and reliability of the simulations. In this paper we propose several new polygonal refinement strategies and numerically investigate the quality of the meshes generated by an adaptive mesh refinement process, as well as optimal rates of convergence with respect to the number of degrees of freedom. Among the several possible problems in which these strategies can be applied, here we have considered a geometrically complex geophysical problem as test problem that naturally yields to a polygonal mesh and tackled it by the Virtual Element Method. All the adaptive strategies here proposed, but the "Trace Direction strategy", can be applied to any problem for which a polygonal element method can be useful and any numerical method based on polygonal elements and can generate good quality isotropic mesh elements.