Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. (English) Zbl 1159.76010

Çeşmelioğlu, A.; Rivière, B.

J. Numer. Math. 16, No. 4, 249-280 (2008).

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:

76D05	Navier-Stokes equations for incompressible viscous fluids
76S05	Flows in porous media; filtration; seepage
35Q30	Navier-Stokes equations
76M20	Finite difference methods applied to problems in fluid mechanics
76M10	Finite element methods applied to problems in fluid mechanics

Keywords:

existence; Beaver-Joseph-Saffman law; weak solution; finite element method; Crank-Nicolson discretization; first-order backward Euler method

References:

[1]	DOI: 10.1007/s10596-007-9043-0 · Zbl 1186.76660
[2]	DOI: 10.1007/BF02576171 · Zbl 0593.76039
[3]	DOI: 10.1017/S0022112067001375
[4]	DOI: 10.1016/j.cam.2005.11.022 · Zbl 1101.76032
[5]	DOI: 10.1016/S0168-9274(02)00125-3 · Zbl 1023.76048
[6]	DOI: 10.1137/06065091X · Zbl 1139.76030
[7]	DOI: 10.1007/s11242-005-1457-3
[8]	DOI: 10.1137/S0036142901392766 · Zbl 1037.76014
[9]	DOI: 10.1137/050637820
[10]	DOI: 10.1007/s10915-004-4147-3 · Zbl 1065.76143
[11]	DOI: 10.1137/S0036142903427640 · Zbl 1084.35063
[12]	Saffman P., Stud. Appl. Math. 50 pp 292– (1971)