18.01.2018 Type Seminar in the framework of the DK Winter Workshop and SFB Internal Meeting Time 09:50 - 10:35 Speaker Lorenzo Mascotto (University of Vienna) Title The $hp$ version of the virtual element method Abstract The virtual element method (VEM) is a recent generalization of the finite element method (FEM) to polygonal/polyhedral meshes, which enable the possibility of dealing with hanging nodes, non-matching grids and easy meshing of geometric data features. The reason for which the method is called "virtual" is that local spaces contain, in addition to polynomials, other functions that are not known in closed-form but which allow to build global $H^1$ conforming spaces. In the standard framework of FEM for the approximation of elliptic Partial Differential Equations (PDEs) on polygonal domains, it is possible to prove that the "output of the method" converges to the exact solution either by considering a sequence of triangular-quadrilateral/tetrahedral-hexaedral meshes with meshsize function converging to zero and by keeping the polynomial degree fixed ($h$-version) or by keeping fixed an underlying mesh and by increasing the polynomial degree ($p$-version). Whenever the solution of the target PDE is analytic, then the $p$-version turns out to be better than the $h$-version, since the decay of the energy error employing the former version of the method is exponential (in terms of $p$), whereas employing the latter is only algebraic (in terms of $h$). At any rate, solutions of elliptic PDEs on a polygonal domain are not analytic in general but typically have singular behaviour at the corners of the domain. Thus, exponential convergence is not anymore valid when employing the $p$-version of the method, but yet can be recovered by combining $h$ and $p$ refinements ($hp$-version). In the present talk, we firstly review basic principles of $hp$ FEM and of standard VEM for the approximation of Poisson problem on polygonal domains. Secondly, we explain how and why to dovetail these two technologies together. Finally, we present some recent advancements of $hp$ VEM.