The objective is to simulate challenging Discrete Fracture Matrix models (DFM) using an efficient vertex approximate gradient scheme as the figure shows. This strategy turns out to be more robust than the standard phase-by-phase approach and it allows reasonable large time steps even the capillary pressure contrasts are important.

Work in collaboration with Konstantin Brenner & Roland Masson

This test exhibits the behavior of Two Point Flux Approximation Hybrid Upwinding (TPFA HU) scheme for the resolution of the 2D-DFM system. The scheme involves physical primary unknowns and includes an nonlinear interface problem. We succeeded to prove the convergence of the developed finite volume scheme and of the discrete transmission condition.

Work in collaboration with Konstantin Brenner, Jérôme Droniou & Roland Masson

This example concerns the numerical cell density which slowly diffuses (no blow-up) towards the chemoattractant regions in a homogeneous way due to the problem's degeneracy.

In this situation, we consider an anisotropic rotated diffusivity that we can excellently recover at the numerical level without any undershoots nor overshoots. It is clearly seen that the cell density obeys the prescribed preferential axes.

When the mesh is severely distorted, we still arrive to obtain admissible cell density profile.

These works have been the object of collaboration with Moustafa Ibrahim & Mazen Saad.