Coupling Stokes and Darcy equations. (English) Zbl 1134.76033

Summary: We study an interface problem between a fluid flow governed by Stokes equations, and a flow in a porous medium, governed by Darcy equations. We consider a weak formulation of the coupled problem which allows to use classical Stokes finite elements in the fluid domain, and standard continuous piecewise polynomials in the porous medium domain. Meshes do not need to match at the interface. The formulation of Stokes equations is standard, while a Galerkin least-squares formulation is used for a mixed form of Darcy equations. We prove the well-posedness of the coupled problem for this formulation and the convergence for some finite element approximations. We also give a two-dimensional numerical example.


76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76S05 Flows in porous media; filtration; seepage
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[1] T. Arbogast, D.S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, preprint, 2004 · Zbl 1186.76660
[2] Beavers, G.; Joseph, D., Boundary conditions at a naturally impermeable wall, J. fluid. mech., 30, 197-207, (1967)
[3] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer New York · Zbl 0788.73002
[4] Brezzi, F.; Fortin, M.; Marini, L.D., Mixed finite element methods with continuous stresses, Math. models methods appl. sci., 3, 275-287, (1993) · Zbl 0774.73066
[5] Burman, E.; Hansbo, P., Stabilized crouzeix – raviart element for the darcy – stokes problem, Numer. meth. part. diff. eq., 21, 986-997, (2005) · Zbl 1077.76037
[6] M. Discacciati, Domain decomposition methods for the coupling of surface and groundwater flows, PhD thesis, École Polytechnique Fédérale de Lausanne, 2004
[7] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. num. math., 43, 57-74, (2002) · Zbl 1023.76048
[8] Discacciati, M.; Quarteroni, A., Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, () · Zbl 1254.76051
[9] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comput., 34, 495-508, (1989) · Zbl 0669.76051
[10] Ern, A.; Guermond, J.-L., Theory and practice of finite elements, (2004), Springer New York
[11] Fortin, M., Old and new finite elements for incompressible flows, Int. J. numer. meth. fluids, 1, 347-364, (1981) · Zbl 0467.76030
[12] Jäger, W.; Mikelić, A., On the interface boundary condition of Beavers, Joseph and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088
[13] Layton, W.J.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. numer. anal., 40, 2195-2218, (2003) · Zbl 1037.76014
[14] Lions, J.-L.; Magenes, E., Problèmes aux limites non homogènes et applications, vol. 1, (1968), Dunod Paris · Zbl 0165.10801
[15] Masud, A.; Hughes, T.J.R., Stabilized mixed finite element method for Darcy flow, Comput. meth. appl. mech. eng., 191, 4341-4370, (2002) · Zbl 1015.76047
[16] Nitsche, J.A., Über ein variationsprinzip zur Lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. math. sem. univ. Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[17] Pierre, R., Simple \(C^0\) approximations for the computation of incompressible flows, Comput. meth. appl. mech. eng., 68, 205-227, (1988) · Zbl 0628.76040
[18] Rivière, B., Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. scient. comp., 22, 479-500, (2005) · Zbl 1065.76143
[19] Rivière, B.; Yotov, I., Locally conservative coupling of Stokes and Darcy flows, SIAM J. numer. anal., 42, 1959-1977, (2005) · Zbl 1084.35063
[20] Roberts, J.E.; Thomas, J.-M., Mixed and hybrid methods, (), 523-639 · Zbl 0875.65090
[21] Saffman, P.G., On the boundary condition at the interface of a porous medium, Stud. appl. math., 1, 93-101, (1971) · Zbl 0271.76080
[22] Urquiza, J.M.; N’Dri, D.; Garon, A.; Delfour, M.C., A numerical study of primal mixed finite element approximations of Darcy equations, Comm. num. meth. eng., 22, 901-915, (2006) · Zbl 1125.76047
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