Local and Parallel Finite Element Methods Based on Two-grid Discretizations for a Transient Coupled Navier-Stokes/Darcy Model

In this paper, some local and parallel finite element methods based on two-grid methods are presented for the non-stationary Navier-Stokes/Darcy model. Based on two-grid methods for spatial discretizations, both semi-discrete scheme and fully-discrete scheme with backward Euler method for the temporal discretization are proposed. Some local a priori estimate, which is crucial for the theoretical analysis, is obtained. The motivation of these local and parallel methods is that by utilizing decoupled method based on interface approximation via temporal extrapolation, low frequency could be obtained on the whole domain with a coarse grid, then solve some residual equations on some overlapped subdomains with a finer gird by some local and parallel procedures at each time step to catch high frequency. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations on the previous time step. To overcome the global discontinuity of the numerical solution generated by the local and parallel finite element algorithms, a new parallel algorithm based on the partition of unity is developed. In the end, some numerical experiments are constructed to prove the effectiveness of our algorithms.