A Newton type linearization based two grid method for coupling fluid flow with porous media flow

In this paper, we propose a two-grid finite element method for solving the mixed Navier-Stokes/Darcy model with the Beavers-Joseph-Saffman interface condition. After solving a coupled nonlinear problem on a coarse grid, we sequentially solve decoupled and linearized subproblems on a fine grid and then correct the solution on the same grid. Compared with the existing work on the two-grid methods for the coupled model, our two-grid method allows a much higher order scaling between the coarse grid size H and the fine grid size h. Specifically, if a k-th order discretization is applied, by using h=H2k+1k for k=1,2 and h=Hk+1k−1 for k≥3, the final step two-grid solution errors in the energy norm are still optimal. Numerical experiments are also given to confirm the theoretical analysis.