Electron. J. Diff. Equ., Vol. 2012 (2012), No. 204, pp. 1-22.

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

Rabah-Hacene Bellout

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.

An addendum was attached on September 16, 2013. It addresses the concerns of a reader about the results being incorrect. See the last page of this article.

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Rabah-Hacene Bellout
Faculté de Mathématiques,
Université des Sciences et Technologies
Houari Boumediene, Algiers, Algeria
email: rbellout@usthb.dz

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