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L. Mascotto:
"The hp version of the Virtual Element Method";
Supervisor, Reviewer: L. Beirao da Veiga, J. Melenk, A. Chernov; Institut für Analysis und Scientific Computing, 2021; oral examination: 2018-02-26.

Chapter 1
Introduction
1.1 Aim of the thesis
The interest in Galerkin methods for the approximation of solutions to partial differential equations (PDEs in short) based on polytopal (i.e. polygonal, polyhedral, . . . ) meshes has recently
grown, due to the high-flexibility that such meshes allow. In fact, employing polytopal meshes automatically includes the possibility of using nonconvex elements, hanging nodes (enabling natural
handling of interface problems with nonmatching grids), easy construction of adaptive meshes and efficient approximations of geometric data features.
We provide here an (incomplete and short) list of polytopal methods: hybrid high-order methods (HHO) [62], mimetic finite difference (MFD) [34, 50], hybrid discontinuous Galerkin methods (HDGM) [60], polygonal finite element method (PFEM) [69, 86, 99], polygonal discontinu-
ous Galerkin methods (DG-FEM) [53, 104], boundary element method-based FEM (BEM-based FEM) [92], weak Galerkin methods (WGM) [103].
The virtual element method (VEM in short) is an alternative (and among the most successful)approach enabling computation on polygonal (polyhedral in 3D) meshes [25, 30]. It is based on
globally continuous discretization spaces that generally consist locally of Trefftz-like functions.
More precisely, the key idea of the VEM is that trial and test spaces consists of functions that are solutions to local PDE problems in each element. Since these local problems do not admit closed-form solutions, the bilinear form, and thereby the entries of the stiffness matrix, are not computable in general. The computable version involves an approximate discrete bilinear form consisting of two additive parts: the first one involves local projections on polynomial spaces, the second one is a computable stabilizing bilinear form. We emphasize that the approximated discrete bilinear form can be evaluated without explicit knowledge of local basis functions in the interior of
the polygonal element: an indirect description via the associated set of internal degrees of freedom suffices.