In this work, an optimization based approach presented in [5, 6, 7] for Discrete Fracture Network simulations is coupled with the Virtual Element Method (VEM) for the space discretization of the underlying Darcy law. The great flexibility of the VEM in handling rather general polygonal elements allows, in a natural way, for an effective description of irregular solutions starting from an arbitrary triangulation, which is built independently of the mesh on other fractures. Only partial conformity is in fact obtained with this approach. Numerical results performed on several DFN configurations confirm the viability and efficiency of the resulting method.

A new approach for solving flow in Discrete Fracture Networks (DFN) is developed in this work by means of the Virtual Element Method. Taking advantage of the features of the VEM, we obtain global conformity of all fracture meshes while preserving a fracture-independent meshing process. This new approach is based on a generalization of globally conforming Finite Elements for polygonal meshes that avoids complications arising from the meshing process. The approach is robust enough to treat many DFNs with a large number of fractures with arbitrary positions and orientations, as shown by the simulations. Higher order Virtual Element spaces are also included in the implementation with the corresponding convergence results and accuracy aspects.

The most challenging issue in performing underground flow simulations in Discrete Fracture Networks (DFN), is to effectively tackle the geometrical difficulties of the problem. In this work we put forward a new application of the Virtual Element Method combined with the Mortar method for domain decomposition: we exploit the flexibility of the VEM in handling polygonal meshes in order to easily construct meshes conforming to the traces on each fracture, and we resort to the mortar approach in order to "weakly" impose continuity of the solution on intersecting fractures. The resulting method replaces the need for matching grids between fractures, so that the meshing process can be performed independently for each fracture. Numerical results show optimal convergence and robustness in handling very complex geometries.

In the framework of the discretization of advection-diffusion problems by means of the Virtual Element Method, we consider stabilization issues. Herein, stabilization is pursued by adding a consistent SUPG-like term. For this approach we prove optimal rates of convergence. Numerical results clearly show the stabilizing effect of the method up to very large Péclet numbers and are in very good agreement with the expected rate of convergence.