Efficient Galerkin-Mixed FEMs for Incompressible Miscible Flow in Porous Media

Abstract

The paper focuses on numerical study of the incompressible miscible flow in porous media. The proposed algorithm is based on a fully decoupled and linearized scheme in the temporal direction, classical Galerkin-mixed approximations in the FE space ($V^r_h$, $S^{r-1}_h$ × $H^{r-1}_h$) ($r$ ≥ 1) in the spatial direction and a post-processing technique for the velocity/pressure, where $V^r_h$ and $S^{r-1}_h$ × $H^{r-1}_h$ denotes the standard $C^0$ Lagrange FE and the Raviart-Thomas FE spaces, respectively. The decoupled and linearized Galerkin-mixed FEM enjoys many advantages over existing methods. At each time step, the method only requires solving two linear systems for the concentration and velocity/pressure. Analysis in our recent work [37] shows that the classical Galerkin-mixed method provides the optimal accuracy $O$($h^{r+1}$) for the numerical concentration in $L^2$-norm, instead of $O$($h^r$) as shown in previous analysis. A new numerical velocity/pressure of the same order accuracy as the concentration can be obtained by the post-processing in the proposed algorithm. Extensive numerical experiments in both two- and three-dimensional spaces, including smooth and non-smooth problems, are presented to illustrate the accuracy and stability of the algorithm. Our numerical results show that the one-order lower approximation to the velocity/pressure does not influence the accuracy of the numerical concentration, which is more important in applications.