Advanced numerical techniques for the simulation of flows in fractured media

The thesis deals with theoretical and applicative aspects of some innovative numerical techniques for the simulation of the flow in Discrete Fracture Networks (DFN). In particular, the recently developed Virtual Element Method (VEM) is considered. A VEM-SUPG stabilized formulation for advection-diffusion problems is defined and studied theoretically and numerically, as well as a residual a posteriori error estimate which does not include any term depending on the VEM stabilization form. Regarding DFN flow simulations, an approach based on Virtual Elements and standard domain decomposition techniques such as Mortar methods is introduced and studied, also in combination with the use of orthogonal polynomials to avoid numerical instabilities that arise when computing polynomial projections on very badly shaped elements. Finally, we consider a constrained optimization formulation of the problem of computing the flow in DFNs and we develop a residual based a posteriori error estimate that contains non standard terms related to the geometrical non-conformity of the mesh on each fracture to the intersections between them.