Non-matching and polytopic finite element techniques with applications to multilevel methods

The numerical approximation of partial differential equations (PDEs) involves the challenge of handling complex geometries and their discretization. In this thesis, we focus on different computational aspects that can be applied in a large variety of scientific and industrial contexts. The first part of the thesis delves into the complexities of managing non-matching grids. Indeed, a common feature to many multi-physics problems is the need for the transfer of data or information between different meshes. We first consider the problem of computing coupling matrices, which require the integration of functions defined on different, arbitrarily overlapped, meshes. We discuss the relevant implementation details and provide a comparison between different unfitted methods. Additionally, we note that the transfer of discrete fields plays a crucial role in several other contexts, e.g. within multilevel methods. Motivated by the excellent properties of multilevel solvers and the performance gain given by matrix-free methodologies, we present a parallel and matrix-free implementation of the non-nested multigrid method. It allows for completely independent and distributed multigrid levels, thereby increasing flexibility on the choice of the hierarchy, while avoiding the explicit assembly of sparse matrices. The second part is devoted to the task of implementing efficient agglomeration procedures, within the polytopic discontinuous Galerkin setting. We develop a novel and efficient approach to perform grid agglomeration using spatial data structures, and validate its robustness and performances also in the memory-distributed setting. Such a coarsening strategy is particularly appealing for multigrid methodologies, as it can deliver a hierarchy of nested grids out of a given geometry. We successfully exploited such versatility in the realistic setting of cardiac electrophysiology, by using our agglomeration procedure to build a multilevel preconditioner for a DG discretization of the monodomain problem. Finally, a proof of the convergence properties of our multilevel strategy is presented. Many results presented in this thesis are also software contributions integrated into the C++ finite element library DEAL.II.