Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Gatica, Gabriel; Oyarzúa, Ricardo; Sayas, Francisco-Javier

doi:10.1090/S0025-5718-2011-02466-X

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem
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by Gabriel N. Gatica, Ricardo Oyarzúa and Francisco-Javier Sayas PDF

Math. Comp. 80 (2011), 1911-1948 Request permission

Abstract:

In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The fully-mixed concept employed here refers to the fact that we consider dual-mixed formulations in both media, which means that the main unknowns are given by the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. We apply the Fredholm and Babuška-Brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous and discrete formulations. In particular, we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates. A feasible choice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and confirming the theoretical rate of convergence, are provided.

References

Abdellatif Agouzal and Jean-Marie Thomas, An extension theorem for equilibrium finite elements spaces, Japan J. Indust. Appl. Math. 13 (1996), no. 2, 257–266. MR 1394627, DOI 10.1007/BF03167247
Todd Arbogast and Dana S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, Comput. Geosci. 11 (2007), no. 3, 207–218. MR 2344200, DOI 10.1007/s10596-007-9043-0
Douglas N. Arnold, Jim Douglas Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. MR 761879, DOI 10.1007/BF01379659
Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
Ivo Babuška and Gabriel N. Gatica, On the mixed finite element method with Lagrange multipliers, Numer. Methods Partial Differential Equations 19 (2003), no. 2, 192–210. MR 1958060, DOI 10.1002/num.10040
G. Beavers and D. Joseph, Boundary conditions at a naturally impermeable wall. Journal of Fluid Mechanics, vol. 30, pp. 197-207, (1967).
Christine Bernardi, Frédéric Hecht, and Olivier Pironneau, Coupling Darcy and Stokes equations for porous media with cracks, M2AN Math. Model. Numer. Anal. 39 (2005), no. 1, 7–35. MR 2136198, DOI 10.1051/m2an:2005007
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Erik Burman and Peter Hansbo, Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem, Numer. Methods Partial Differential Equations 21 (2005), no. 5, 986–997. MR 2154230, DOI 10.1002/num.20076
Erik Burman and Peter Hansbo, A unified stabilized method for Stokes’ and Darcy’s equations, J. Comput. Appl. Math. 198 (2007), no. 1, 35–51. MR 2250387, DOI 10.1016/j.cam.2005.11.022
M.R. Correa, Stabilized Finite Element Methods for Darcy and Coupled Stokes-Darcy Flows. D.Sc. Thesis, LNCC, Petrópolis, Rio de Janeiro, Brasil (in Portuguese), (2006).
M.R. Correa and A.F.D. Loula, A unified mixed formulation naturally coupling Stokes and Darcy flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 33-36, pp. 2710-2722, (2009).
M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne, 2004.
Marco Discacciati, Edie Miglio, and Alfio Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math. 43 (2002), no. 1-2, 57–74. 19th Dundee Biennial Conference on Numerical Analysis (2001). MR 1936102, DOI 10.1016/S0168-9274(02)00125-3
Víctor Domínguez and Francisco-Javier Sayas, Stability of discrete liftings, C. R. Math. Acad. Sci. Paris 337 (2003), no. 12, 805–808 (English, with English and French summaries). MR 2033124, DOI 10.1016/j.crma.2003.10.025
V. J. Ervin, E. W. Jenkins, and S. Sun, Coupled generalized nonlinear Stokes flow with flow through a porous medium, SIAM J. Numer. Anal. 47 (2009), no. 2, 929–952. MR 2485439, DOI 10.1137/070708354
Avner Friedman, Foundations of modern analysis, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100
Juan Galvis and Marcus Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations, Electron. Trans. Numer. Anal. 26 (2007), 350–384. MR 2391227
Gabriel N. Gatica, Norbert Heuer, and Salim Meddahi, On the numerical analysis of nonlinear twofold saddle point problems, IMA J. Numer. Anal. 23 (2003), no. 2, 301–330. MR 1975268, DOI 10.1093/imanum/23.2.301
Gabriel N. Gatica, Salim Meddahi, and Ricardo Oyarzúa, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow, IMA J. Numer. Anal. 29 (2009), no. 1, 86–108. MR 2470941, DOI 10.1093/imanum/drm049
G.N. Gatica, R. Oyarzúa, and F.-J. Sayas, Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem. Numerical Methods for Partial Differential Equations DOI 10.1002/num.
Gabriel N. Gatica and Francisco-Javier Sayas, Characterizing the inf-sup condition on product spaces, Numer. Math. 109 (2008), no. 2, 209–231. MR 2385652, DOI 10.1007/s00211-008-0140-3
V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867, DOI 10.1007/BFb0063453
P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
Willi Jäger and Andro Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math. 60 (2000), no. 4, 1111–1127. MR 1760028, DOI 10.1137/S003613999833678X
Trygve Karper, Kent-Andre Mardal, and Ragnar Winther, Unified finite element discretizations of coupled Darcy-Stokes flow, Numer. Methods Partial Differential Equations 25 (2009), no. 2, 311–326. MR 2483769, DOI 10.1002/num.20349
William J. Layton, Friedhelm Schieweck, and Ivan Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 40 (2002), no. 6, 2195–2218 (2003). MR 1974181, DOI 10.1137/S0036142901392766
Arif Masud, A stabilized mixed finite element method for Darcy-Stokes flow, Internat. J. Numer. Methods Fluids 54 (2007), no. 6-8, 665–681. MR 2333004, DOI 10.1002/fld.1508
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729, DOI 10.1007/978-3-540-85268-1
María-Luisa Rapún and Francisco-Javier Sayas, Boundary integral approximation of a heat-diffusion problem in time-harmonic regime, Numer. Algorithms 41 (2006), no. 2, 127–160. MR 2222969, DOI 10.1007/s11075-005-9002-6
Béatrice Rivière, Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. Sci. Comput. 22/23 (2005), 479–500. MR 2142206, DOI 10.1007/s10915-004-4147-3
Béatrice Rivière and Ivan Yotov, Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal. 42 (2005), no. 5, 1959–1977. MR 2139232, DOI 10.1137/S0036142903427640
Hongxing Rui and Ran Zhang, A unified stabilized mixed finite element method for coupling Stokes and Darcy flows, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 33-36, 2692–2699. MR 2532369, DOI 10.1016/j.cma.2009.03.011
P. Saffman, On the boundary condition at the surface of a porous media. Studies in Applied Mathematics, vol. 50, pp. 93-101, (1971).
J. M. Urquiza, D. N’Dri, A. Garon, and M. C. Delfour, Coupling Stokes and Darcy equations, Appl. Numer. Math. 58 (2008), no. 5, 525–538. MR 2407730, DOI 10.1016/j.apnum.2006.12.006
Xiaoping Xie, Jinchao Xu, and Guangri Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math. 26 (2008), no. 3, 437–455. MR 2421892