Finite element method for the stationary dual-porosity Navier-Stokes system with Beavers-Joseph interface conditions.

This work proposes a coupled finite element method for solving the stationary dual-porosity Navier-Stokes system with the Beavers-Joseph interface condition and the normal balance forces condition without the additional inertial term of 12(uc⋅uc). The major mathematical difficulty of this system lies in the indefinite leading order contributed by the Beavers-Joseph condition to the total energy. The remedy is to introduce a proper re-scaling factor for the coupled system and transfer the original problem into a new space for discussion. An a priori estimate as well as the existence and local uniqueness are established under suitable assumptions of physical parameters (such as exchange coefficient α, the fluid density ρ) and the large sufficient fluid viscosity ν. Error estimates and convergence of the coupled finite element approximation are obtained. Finally, two numerical experiments are provided to illustrate the robustness of the proposed algorithm and the features of the coupled system.