The virtual element method on anisotropic polygonal discretizations

In recent years, the numerical treatment of boundary value problems with the help of polygonal and polyhedral discretization techniques has received a lot of attention within several disciplines. Due to the general element shapes an enormous flexibility is gained and can be exploited, for instance, in adaptive mesh refinement strategies. The Virtual Element Method (VEM) is one of the new promising approaches applicable on general meshes. Although polygonal element shapes may be highly adapted, the analysis relies on isotropic elements which must not be very stretched. But, such anisotropic element shapes have a high potential in the discretization of interior and boundary layers. Recent results on anisotropic polygonal meshes are reviewed and the Virtual Element Method is applied on layer adapted meshes containing isotropic and anisotropic polygonal elements.