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Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. (English) Zbl 1159.76010
Summary: 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 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.
MSC:
76D05Navier-Stokes equations (fluid dynamics)
76S05Flows in porous media; filtration; seepage
35Q30Stokes and Navier-Stokes equations
76M20Finite difference methods (fluid mechanics)
76M10Finite element methods (fluid mechanics)