Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 204, pp. 1-22.

Title: Numerical approximation for a degenerate parabolic-elliptic 
       system modeling flows in porous media
 
Author: Rabah-Hacene Bellout (Univ. des Sciences et Tech., Algiers, Algeria}

Abstract: 
 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
 $ 0\leq 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.

Submitted August 1, 2012. Published November 24, 2012.
Math Subject Classifications: 76S05, 35K65, 35B65, 65M06.
Key Words: Mixed finite element methods; finite volume methods; 
           porous media.