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On the solution of the coupled Navier-Stokes and Darcy equations. (English) Zbl 1230.76023
Summary: This paper introduces and analyzes two mathematical models for coupling the incompressible Navier-Stokes equations with the porous media flow equations. A numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium, is proposed. Existence and uniqueness results under small data condition of the numerical solution are proved. Optimal a priori error estimates are derived. Numerical examples comparing the two models under varying physical parameters are provided.
MSC:
76M10Finite element methods (fluid mechanics)
76D05Navier-Stokes equations (fluid dynamics)
76S05Flows in porous media; filtration; seepage
76M30Variational methods (fluid mechanics)
References:
[1]Adams, R.: Sobolev spaces, (1975)
[2]Arnold, D. N.: An interior penalty finite element method with discontinuous elements, SIAM J. Numer. anal. 19, 742-760 (1982) · Zbl 0482.65060 · doi:10.1137/0719052
[3]Arnold, D. N.; Brezzi, F.; Fortin, M.: A stable finite element for the Stokes equations, Calcolo 21, 337-344 (1982) · Zbl 0593.76039 · doi:10.1007/BF02576171
[4]Beavers, G. S.; Joseph, D. D.: Boundary conditions at a naturally impermeable wall, J. fluid mech. 30, 197-207 (1967)
[5]Burman, E.; Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations, J. comput. Appl. math. 198, No. 1, 35-51 (2007) · Zbl 1101.76032 · doi:10.1016/j.cam.2005.11.022
[6]Cesmelioglu, A.; Rivière, B.: Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow, J. sci. Comput. 40, 115-140 (2009) · Zbl 1203.76078 · doi:10.1007/s10915-009-9274-4
[7]Ciarlet, P.: The finite element method for elliptic problems, (1978)
[8]P. Chidyagwai, B. Rivière, Analysis of two mathematical models for the coupled Navier – Stokes/Darcy problem, Technical Report, Computational and Applied Mathematics Department, Rice University, TR09-14, 2009.
[9]Dawson, C.; Sun, S.; Wheeler, M. F.: Compatible algorithms for coupled flow and transport, Comput. meth. Appl. mech. Engrg. 193, 2565-2580 (2004) · Zbl 1067.76565 · doi:10.1016/j.cma.2003.12.059
[10]M. Discacciati, Domain decomposition methods for the coupling of surface and groundwater flows, PhD Thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2004.
[11]Discacciati, M.; Miglio, E.; Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows, Appl. numer. Math. 43, 57-74 (2001) · Zbl 1023.76048 · doi:10.1016/S0168-9274(02)00125-3
[12]Discacciati, M.; Quarteroni, A.: Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, Numerical analysis and advanced applications – ENUMATH 2001, 3-20 (2003)
[13]Discacciati, M.; Quarteroni, A.; Valli, A.: Robin – Robin domain decomposition methods for the Stokes – Darcy coupling, SIAM J. Numer. anal. 45, No. 3, 1246-1268 (2007) · Zbl 1139.76030 · doi:10.1137/06065091X
[14]Epshteyn, Y.; Rivière, B.: Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J. comput. Appl. math. 206, 843-872 (2007) · Zbl 1141.65078 · doi:10.1016/j.cam.2006.08.029
[15]Girault, V.; Raviart, P. -A.: Finite element methods for Navier – Stokes equations: theory and algorithms, Finite element methods for Navier – Stokes equations: theory and algorithms 5 (1986) · Zbl 0585.65077
[16]Girault, V.; Rivière, B.: DG approximation of coupled Navier – Stokes and Darcy equations by beaver – Joseph – Saffman interface condition, SIAM J. Numer. anal. 47, 2052-2089 (2009)
[17]Girault, V.; Rivière, B.; Wheeler, M.: A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier – Stokes problems, Math. comput. 74, 53-84 (2004) · Zbl 1057.35029 · doi:10.1090/S0025-5718-04-01652-7
[18]Hanspal, N. S.; Waghode, A. N.; Nassehi, V.; Wakeman, R. J.: Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations, Transport porous med. 64, No. 1, 1573-1634 (2006)
[19]Heywood, J. G.; Rannacher, R.; Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier – Stokes equations, Int. J. Numer. methods fluids 22, 325-352 (1996) · Zbl 0863.76016 · doi:10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
[20]Hood, P.; Taylor, C.: A numerical solution of the Navier – Stokes equations using the finite element technique, Comput. fluids 1, 73-100 (1973) · Zbl 0328.76020 · doi:10.1016/0045-7930(73)90027-3
[21]Houston, P.; Schwab, C.; Süli, E.: Discontinuous hp-finite element methods for advection – diffusion problems, SIAM J. Numer. anal. 39, No. 6, 2133-2163 (2002) · Zbl 1015.65067 · doi:10.1137/S0036142900374111
[22]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 · doi:10.1137/S003613999833678X
[23]Layton, W. J.; Schieweck, F.; Yotov, I.: Coupling fluid flow with porous media flow, SIAM J. Numer. anal. 40, No. 6, 2195-2218 (2003) · Zbl 1037.76014 · doi:10.1137/S0036142901392766
[24]Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. sci. Comput. 22, 479-500 (2005) · Zbl 1065.76143 · doi:10.1007/s10915-004-4147-3
[25]Rivière, B.: Analysis of a multi-numerics/multi-physics problem, Numer. math. Adv. appl., 726-735 (2004) · Zbl 1216.76033
[26]Rivière, B.; Wheeler, M. F.; Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. anal. 39, No. 3, 902-931 (2001) · Zbl 1010.65045 · doi:10.1137/S003614290037174X
[27]Rivière, B.; Yotov, I.: Locally conservative coupling of Stokes and Darcy flow, SIAM J. Numer. anal. 42, 1959-1977 (2005) · Zbl 1084.35063 · doi:10.1137/S0036142903427640
[28]Saffman, P.: On the boundary condition at the surface of a porous media, Stud. appl. Math. 50, 292-315 (1971) · Zbl 0271.76080
[29]Wheeler, M. F.: An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. anal. 15, No. 1, 152-161 (1978) · Zbl 0384.65058 · doi:10.1137/0715010