Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow

This paper formulates and analyzes a weak solution to the coupling of time-dependent Navier-Stokes flow with Darcy flow under certain boundary conditions, one of them being the Beaver Joseph-Saffman law on the interface. Existence and a priori estimates for the weak solution are shown under additional regularity assumptions. We introduce a fully discrete scheme with the unknowns being the Navier-Stokes velocity, pressure and the Darcy pressure. The scheme we propose is based on a finite element method in space and a Crank-Nicolson discretization in time where we obtain the solution at the first time step using a first order backward Euler method. Convergence of the scheme is obtained and optimal error estimates with respect to the mesh size are derived.