Numerical approximation for a degenerate parabolic-elliptic system modeling flows in porous media

We present a numerical scheme for the approximation of the system of partial differential equations of the Peaceman model for the miscible displacement of one fluid by another in a two dimensional porous medium. In this scheme, the velocity-pressure equations are treated by a mixed finite element discretization using the Raviart-Thomas element, and the concentration equation is approximated by a finite volume discretization using the Upstream scheme, knowing that the Raviart-Thomas element gives good approximations for fluids velocities and that the Upstream scheme is well suited for convection dominated equations. We prove a maximum principle for our approximate concentration more precisely $ 0leq c_h(x,t)leq 1$ a.e. in $Omega_T $ as long as some grid conditions are satisfied - at the difference of Chainais and Droniou [6]who have only observed that their approximate concentration remains in $[0;1]$ (and such is the case for other proposed numerical methods; e.g., [21,22]. Moreover our grid conditions are satisfied even with very large time steps and spatial steps. Finally we prove the consistency of the proposed scheme and thus are assured of convergence. A numerical test is reported.