Two-level Galerkin Mixed Finite Element Method For Darcy-Forchheimer Incompressible Miscible Displacement Problem

In this paper,the two grid mixed finite element method is used to solve the incompress-ible miscible displacement problem in porous media,in which the relationship between velocity and pressure conforms to Darcy Forchheimer law.The model of the problem is as follows (?)The two grid method transforms solving a large nonlinear system on the fine grid into solving a small nonlinear system on the coarse grid and solving a linear system on the fine grid.The iterative method is needed to solve the nonlinear system,but only one Newtonian iteration step is carried out after the transformation into a linear system,which can greatly improve the speed of calculation.In this paper,for the non-linear term |u|u in the Darcy-Forchheimer equation,it takes a lot of time to solve it.Therefore,we choose the two grid method to save the calculation time.We divide the region,the size of coarse mesh is H and the size of fine mesh is h,where h<<H.First,we solve the problem on the coarse grid to obtain the numerical solution.Then,Taylor expansion is carried out for the nonlinear terms on fine grid,and the numerical solutions on coarse grid are used to express the nonlinear terms on fine grid.In this way,the nonlinear terms on the fine grid are converted into linear terms,and the solution of the nonlinear system is converted into a linear system,thus greatly reducing the operation time.In this paper,we use piecewise constant to approximate velocity and use piecewise continuous linear function to approximate pressure and concentration,and obtain the error estimates on coarse and fine grids.The numerical experimental results accord with our theoretical derivation,which proves the correctness of the results...