Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed 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 mixed finite element method with characteristics for the
concentration equation. To linearize the mixed-method equations, we use a two-grid
algorithm based on the Newton iteration method for this full discrete scheme problems.
First, we solve the original nonlinear equations on the coarse grid, then, we
solve the linearized problem on the fine grid used Newton iteration once. It is shown
that the coarse grid can be much coarser than the fine grid and achieve asymptotically
optimal approximation as long as the mesh sizes satisfy $h=H^2$ in this paper. Finally,
numerical experiment indicates that two-grid algorithm is very effective.