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.