Positive vertex finite volume schemes for degenerate parabolic equations arising in porous media flow models

My talk will be centered on the control volume finite element method for the approximation of solutions to degenerate convection-diffusion equations. The main point consists in designing a discretization which allows to ensure both the positivity and the coercivity of the numerical scheme. The idea is essentially based on appropriate upwind approximations of the fluxes. A particular emphasize is placed on the convergence of the method while the discrete solution stays physically admissible. Taking also into account accuracy quests, I will present several propositions that retain the mentioned ingredients. Moreover, the approach is successfully applied to a simple diffusion equation. Numerical tests are presented in order to validate the expected results. Next, I will extend the methodology to a challenging system modeling compressible flows in porous media. Finally, I will conclude my presentation with some perspectives on the topic.