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


Authors: Gabriel N. Gatica, Ricardo Oyarzúa and Francisco-Javier Sayas
Journal: Math. Comp. 80 (2011), 1911-1948
MSC (2010): Primary 65N15, 65N30, 74F10, 74S05
DOI: https://doi.org/10.1090/S0025-5718-2011-02466-X
Published electronically: February 14, 2011
MathSciNet review: 2813344
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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.


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  • 1. A. AGOUZAL AND J.-M. THOMAS, An extension theorem for equilibrium finite elements spaces. Japan Journal of Industrial and Applied Mathematics, vol. 13, 2, pp. 257-266, (1996). MR 1394627 (97g:65220)
  • 2. T. ARBOGAST AND D.S. BRUNSON, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Computational Geosciences, vol. 11, 3, pp. 207-218, (2007). MR 2344200 (2009b:76155)
  • 3. D.N. ARNOLD, J. DOUGLAS AND CH.P. GUPTA, A family of higher order mixed finite element methods for plane elasticity. Numerische Mathematik, vol. 45, 1, pp. 1-22, (1984). MR 761879 (86a:65112)
  • 4. I. BABUŠSKA AND A.K. AZIZ, Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz (editor), Academic Press, New York, (1972). MR 0421106 (54:9111)
  • 5. I. BABUˇSKA AND G.N. GATICA, On the mixed finite element method with Lagrange multipliers. Numerical Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003). MR 1958060 (2004b:65174)
  • 6. G. BEAVERS AND D. JOSEPH, Boundary conditions at a naturally impermeable wall. Journal of Fluid Mechanics, vol. 30, pp. 197-207, (1967).
  • 7. C. BERNARDI, F. HECHT, AND O. PIRONNEAU, Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 39, 1, pp. 7-35, (2005). MR 2136198 (2006a:76107)
  • 8. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 9. E. BURMAN AND P. HANSBO, Stabilized Crouzeix-Raviart elements for the Darcy-Stokes problem. Numerical Methods for Partial Differential Equations, vol. 21, 5, pp. 986-997, (2005). MR 2154230 (2006i:65190)
  • 10. E. BURMAN AND P. HANSBO, A unified stabilized method for Stokes' and Darcy's equations. Journal of Computational and Applied Mathematics, vol. 198, 1, pp. 35-51, (2007). MR 2250387 (2007i:65076)
  • 11. 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).
  • 12. 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).
  • 13. M. DISCACCIATI, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne, 2004.
  • 14. M. DISCACCIATI, E. MIGLIO, AND A. QUARTERONI, Mathematical and numerical models for coupling surface and groundwater flows. Applied Numerical Mathematics, vol. 43, pp. 57-74, (2002). MR 1936102 (2003h:76087)
  • 15. V. DOM´İNGUEZ AND F.-J. SAYAS, Stability of discrete liftings. Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 337, 12, pp. 805-808, (2003). MR 2033124
  • 16. V.J. ERVIN, E.W. JENKINS, AND S. SUN, Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM Journal on Numerical Analysis, vol. 47, 2, pp. 929-952, (2009). MR 2485439 (2010b:65254)
  • 17. A. FRIEDMAN, Foundations of Modern Analysis. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100 (43:858)
  • 18. J. GALVIS AND M. SARKIS, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electronic Transactions on Numerical Analysis, vol. 26, pp. 350-384, (2007). MR 2391227 (2009a:76120)
  • 19. G.N. GATICA, N. HEUER, AND S. MEDDAHI, On the numerical analysis of nonlinear twofold saddle point problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003). MR 1975268 (2004b:65183)
  • 20. G.N. GATICA, S. MEDDAHI, AND R. OYARZÚA, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009). MR 2470941 (2010b:76118)
  • 21. 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.
  • 22. G.N. GATICA AND F.-J. SAYAS, Characterizing the inf-sup condition on product spaces. Numerische Mathematik, vol. 109, 2, pp. 209-231, (2008). MR 2385652 (2009g:47034)
  • 23. V. GIRAULT AND P.-A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, 749. Springer-Verlag, Berlin-New York, 1979. MR 548867 (83b:65122)
  • 24. P. GRISVARD, Singularities in Boundary Value Problems. Recherches en Mathématiques Appliquées, 22. Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209 (93h:35004)
  • 25. R. HIPTMAIR, Finite elements in computational electromagnetism. Acta Numerica, vol. 11, pp. 237-339, (2002). MR 2009375 (2004k:78028)
  • 26. W. J¨AGER AND M. MIKELIC, On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM Journal on Applied Mathematics, vol. 60, pp. 1111-1127, (2000). MR 1760028 (2001e:76122)
  • 27. T. KARPER, K.-A. MARDAL, AND R. WINTHER, Unified finite element discretizations of coupled Darcy-Stokes flow. Numerical Methods for Partial Differential Equations, vol. 25, 2, pp. 311-326, (2009). MR 2483769 (2010a:65240)
  • 28. W.J. LAYTON, F. SCHIEWECK, AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003). MR 1974181 (2004c:76048)
  • 29. A. MASUD, A stabilized mixed finite element method for Darcy-Stokes flow. International Journal for Numerical Methods in Fluids, vol. 54, 6-8, pp. 665-681, (2008). MR 2333004 (2008b:76120)
  • 30. W. MCLEAN, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 2000. MR 1742312 (2001a:35051)
  • 31. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, Heidelberg, 1994. MR 1299729 (95i:65005)
  • 32. M.L. RAPÚN AND F.-J. SAYAS, Boundary integral approximation of a heat-diffusion problem in time-harmonic regime. Numerical Algorithms, vol. 41, 2, pp. 127-160, (2006). MR 2222969 (2007c:65114)
  • 33. B. RIVIERE, Analysis of a discontinuous finite element method for coupled Stokes and Darcy problems. Journal of Scientific Computing, vol. 22-23, pp. 479-500, (2005). MR 2142206 (2006b:65175)
  • 34. B. RIVIERE AND I. YOTOV, Locally conservative coupling of Stokes and Darcy flows. SIAM Journal on Numerical Analysis, vol. 42, 5, pp. 1959-1977, (2005). MR 2139232 (2006a:76035)
  • 35. H. RUI AND R. ZHANG, A unified stabilized mixed finite element method for coupling Stokes and Darcy flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 33-36, pp. 2692-2699, (2009). MR 2532369 (2010g:76045)
  • 36. P. SAFFMAN, On the boundary condition at the surface of a porous media. Studies in Applied Mathematics, vol. 50, pp. 93-101, (1971).
  • 37. J.M. URQUIZA, D. N'DRI, A. GARON, AND M.C. DELFOUR, Coupling Stokes and Darcy equations. Applied Numerical Mathematics, vol. 58, 5, pp. 525-538, (2008). MR 2407730 (2009a:76053)
  • 38. X. XIE, J. XU, AND G. XUE, Uniformly stable finite element methods for Darcy-Stokes-Brinkman models. Journal of Computational Mathematics, vol. 26, 3, pp. 437-455, (2008). MR 2421892 (2009g:76087)

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Additional Information

Gabriel N. Gatica
Affiliation: CI\raisebox{.75ex}2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: ggatica@ing-mat.udec.cl

Ricardo Oyarzúa
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: royarzua@ing-mat.udec.cl

Francisco-Javier Sayas
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA
Email: fjsayas@math.udel.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02466-X
Received by editor(s): June 3, 2009
Received by editor(s) in revised form: July 12, 2010
Published electronically: February 14, 2011
Additional Notes: The research of the first two authors was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by CI$^{2}$MA, Universidad de Concepción, and by MECESUP project UCO 0713
The third author acknowledges support of MEC/FEDER Project MTM2007–63204 and Gobierno de Aragón (Grupo PDIE). This work was developed while the third author was at the University of Minnesota supported by a Spanish MEC grant PR2007–0016.
Article copyright: © Copyright 2011 American Mathematical Society

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