Superconvergence Of Imcompressible Miscible Displacement Approximated By Mixed Finite Element Methods Combined With A Characteristics Method

The flow and transport of fluids in porous media is of great importance socially and economically in the oil recovery and environmental pollution problem. The incompressible miscible displacement problem is the most traditional one among this kind of problems. It can be modeled by a nonlinear coupled system of two partial differential equations: the pressure-velocity equation, which is elliptic, and the concentration equation, which is parabolic but usually convection-dominated. In this paper, we employ a mixed finite element method to approximate the pressure and the Darcy velocity, and a Galerkin finite element method combined with the MMOC (modified method of characteristics) to approximate the concentration. Especially, the coefficients in convection and diffusion terms of concentration equation only involve the Darcy velocity, therefore, the numerical results of concentration strongly depend on the Darcy velocity. Because of the Darcy velocity has the superconvergent property along the Gauss lines, we obtain its superconvergent numerical results on all of the domain by using post-processing techniques, and then we make it to estimate the coefficients in concentration equation. After strict error analysis, we prove that our algorithm is superconvergence about the pressure step. In the end of this article, we solve a poisson problem with non-homogeneous Neumann boundary condition by mixed element methods and the modified Uzawa's algorithm. The numerical results indicate these two kinds of methods can be have the same superconvergent property along the Gauss lines, and their post-processing are also superconvergence on all of the domain, especially, the latter methods cost fewer computed time than the first. On the base of numerical results, we design a new discrete shceme to solve impressible miscible problem.