zbMATH — the first resource for mathematics

A unified stabilized mixed finite element method for coupling Stokes and Darcy flows. (English) Zbl 1228.76090
Summary: A stabilized mixed finite element method for solving the coupled Stokes and Darcy flows problem is formulated and analyzed. The approach utilizes the same nonconforming Crouzeix-Raviart element discretization on the entire domain. A discrete inf-sup condition and an optimal a priori error estimate are derived. Finally, some numerical examples verifying the theoretical predictions are presented.

MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76D07 Stokes and related (Oseen, etc.) flows
Full Text:
References:
 [1] Beavers, G.S.; Joseph, D.D., Boundary conditions at a naturally permeable wall, J. fluid mech., 30, 197-207, (1967) [2] Saffman, P.G., On the boundary condition at the surface of a porous medium, Stud. appl. math., 50, 93-101, (1971) · Zbl 0271.76080 [3] Gartling, D.K.; Hickox, C.E.; Givler, R.C., Simulation of coupled viscous and porous flow problems, Comp. fluid dyn., 7, 23-48, (1996) · Zbl 0879.76104 [4] Salinger, A.G.; Aris, R.; Derby, J.J., Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains, Int. J. numer. methods fluids, 18, 1185-1209, (1994) · Zbl 0807.76039 [5] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. numer. math., 43, 57-74, (2002) · Zbl 1023.76048 [6] 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 [7] Rivière, B.; Yotov, I., Locally conservative coupling of Stokes and Darcy flows, SIAM J. numer. anal., 42, 1959-1977, (2005) · Zbl 1084.35063 [8] Arbogast, T.; Brunson, D.S., A computational method for approximating a darcy – stokes system governing a vuggy porous medium, Comput. geosci., 11, 207-218, (2007) · Zbl 1186.76660 [9] Fortin, M., Old and new finite elements for incompressible flows, Int. J. numer. meth. fluids, 1, 347-364, (1981) · Zbl 0467.76030 [10] Burman, E.; Hansbo, P., A unified stabilized method for stokes’ and darcy’s equations, J. comput. appl. math., 198, 35-51, (2007) · Zbl 1101.76032 [11] Karper, T.; Mardal, K.-A.; Winther, R., Unified finite element discretizations of coupled darcy – stokes flow, Numer. methods partial differ. eq., 25, 311-326, (2009) · Zbl 1157.76026 [12] Mardal, K.A.; Tai, X.-C.; Winther, R., A robust finite element method for the darcy – stokes flow, SIAM J. numer. anal., 40, 1605-1631, (2002) · Zbl 1037.65120 [13] Burman, E.; Hansbo, P., Stabilized crouzeix – raviart element for the darcy – stokes problem, Numer. methods partial differ. eq., 21, 986-997, (2005) · Zbl 1077.76037 [14] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York · Zbl 0788.73002 [15] Brenner, S.C., Korn’s inequalities for piecewise $$H^1$$ vector fields, Math. comput., 73, 1067-1087, (2003) · Zbl 1055.65118 [16] Crouzeix, M.; Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Rev. française automat. informat. recherche opérationnelle Sér. rouge, 7, 33-75, (1973) · Zbl 0302.65087 [17] Thomée, V., Galerkin finite element methods for parabolic problems, (1997), Springer-Verlag Berlin · Zbl 0884.65097 [18] Girault, V.; Raviart, P.-A., Finite element methods for Navier Stokes equations, (1986), Springer-Verlag Berlin · Zbl 0413.65081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.