Two-grid methods for miscible displacement problem by Galerkin methods and mixed finite-element methods

The miscible displacement problem of one incompressible fluid is modelled by a nonlinear coupled system of two partial differential equations in porous media. One equation is elliptic form for the pressure and the other equation is parabolic form for the concentration of one of the fluids. In the paper, we present an efficient two-grid method for solving the miscible displacement problem by using mixed finite-element method for the approximation of the pressure equation and standard Galerkin method for concentration equation. We linearize the discretized equations based on the idea of Newton iteration in our methods, firstly, we solve an original nonlinear coupling problem on the coarse grid, then solve two linear systems on the fine grid. we obtain the error estimates for the two-grid algorithm, it is shown that coarse space can be extremely coarse and we achieve asymptotically optimal approximation. Moreover, numerical experimentation is given in this paper.