Two-grid method for miscible displacement problem by mixed finite element methods and finite element method of characteristics

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and finite element method with characteristics for the concentration equation. To linearize this full discrete scheme problems, we use two Newton iterations on the fine grid in our methods, with the initial guess coming from the coarse-grid solution. Then, we get the error estimates for the three-step two-grid algorithms. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h14) in the three-step algorithm. Finally, numerical experiment indicates that two-grid algorithm is very effective.