Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics (Communications in Computational Physics, 2020)
The miscible displacement of one incompressible fluid by another in a porousmedium is governed by a system of two equations. One is elliptic form equation forthe pressure and the other is parabolic form equation for the concentration of one ofthe fluids. Since only the velocity and not the pressure appears explicitly in the concentrationequation, we use a mixed finite element method for the approximation ofthe pressure equation and mixed finite element method with characteristics for theconcentration equation. To linearize the mixed-method equations, we use a two-gridalgorithm based on the Newton iteration method for this full discrete scheme problems.First, we solve the original nonlinear equations on the coarse grid, then, wesolve the linearized problem on the fine grid used Newton iteration once. It is shownthat the coarse grid can be much coarser than the fine grid and achieve asymptoticallyoptimal 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.