Local and parallel finite element methods based on two-grid discretizations for a non-stationary coupled Stokes-Darcy model

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for a non-stationary coupled Stokes-Darcy model. Based on two-grid discretizations, a semi-discrete scheme is presented. With backward Euler scheme for the temporal discretization and two-grid discretizations for the spatial discretization, some fully discrete schemes are proposed. The crucial idea is to adopt a decoupling scheme based on interface approximation via temporal extrapolation to approximate the mixed model by utilizing a coarse grid on the whole domain, then solve some residual equations with a finer grid on some overlapped subdomains by some local and parallel procedures at each time step. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations. To reach a global continuous approximation, a new parallel algorithm based on the partition of unity is devised. Some local a priori estimate that is crucial for the theoretical analysis is obtained. Finally, some numerical experiments are conducted to support our theoretical results and demonstrate the computational effectiveness.