A Weak Galerkin Method for the Coupled Darcy-Stokes Problem with a Nonstandard Transmission Condition on the Stokes Boundary

In this paper, we study a coupled problem of Darcy’s law and Stokes equation with nonlinear slip boundary conditions. We derive a variational inequality for the coupled problem in detail. Then we introduce and analyze a weak Galerkin method to solve the coupled problem numerically. Under proper regularity assumptions, we obtain the optimal error estimate $\mathscr{O}(h)$ in newly defined $h$-norms. Finally, we give some numerical results to support the theoretical conclusions.