Coupling finite element and multiscale finite element methods for the non-stationary Stokes-Darcy model

In this paper, we propose and analyze the multiscale finite element method for the non-stationary Stokes-Darcy model, where the permeability coefficient in the Darcy region exhibits multiscale characteristics. Our algorithm involves two main steps: offline, where we solve the multiscale basis functions in the Darcy region via parallel computation; Online, where we decouple the Stokes-Darcy equations in an implicit-explicit manner. One significant feature of the algorithm is that it solves problems on relatively coarse grids while maintaining accuracy, thus significantly reducing computational costs compared to the standard finite element method. Moreover, under the same coarse grid size, it exhibits a significant improvement in accuracy compared to the standard finite element method. Under the assumption that the permeability coefficient is periodic and independent of time, we demonstrate the stability and convergence of the resulting sequential algorithm. Finally, the capability and effectiveness of the algorithm are verified through three numerical experiments, with computational results consistent with theoretical analysis.