Computational Geosciences

, Volume 18, Issue 5, pp 711–728 | Cite as

On classical iterative subdomain methods for the Stokes–Darcy problem



Within classical iterative subdomain methods, the problems in the subdomains are solved alternately by only using data on the interface provided from the other subdomains. Methods of this type for the Stokes–Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if it is used with appropriate parameters in the Robin conditions.


Stokes–Darcy problem Classical iterative subdomain methods Robin boundary conditions Finite element methods Continuous and discontinuous updating strategy Applications from geosciences 


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  1. 1.
    Angot, P.: On the well-posed coupling between free fluid and porous viscous flows. Appl. Math. Lett. 24, 803–810 (2011)CrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Badia, S., Codina, R.: Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47, 1971–2000 (2009)CrossRefGoogle Scholar
  4. 4.
    Burman, E., Hansbo, P.: A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007)CrossRefGoogle Scholar
  5. 5.
    Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, X., Zhao, W.: Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions. SIAMJ. Numer. Anal. 47(6), 4239–4256 (2010)CrossRefGoogle Scholar
  6. 6.
    Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8(1), 1–25 (2010)CrossRefGoogle Scholar
  7. 7.
    Cardenas, M., Wilson, J.: Dunes, turbulent eddies, and interfacial exchange with permeable sediments. Water Resour. Res. 43(08), 412 (2007)Google Scholar
  8. 8.
    Cardenas, M., Wilson, J.: Hydrodynamics of coupled flow above and below a sediment–water interface with triangular bedforms. Adv. Water Resour. 30, 301–313 (2007)CrossRefGoogle Scholar
  9. 9.
    Chen, W., Gunzburger, M., Hua, F., Wang, X.: A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system. SIAM. J. Numer. Anal. 49(3), 1064–1084 (2011)CrossRefGoogle Scholar
  10. 10.
    Davis, T.A.: Algorithm 845 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004)CrossRefGoogle Scholar
  11. 11.
    Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. PhD thesis, École Polytechnique Fédérale de Lausanne (2004)Google Scholar
  12. 12.
    Discacciati, M., Quarteroni, A.: Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22(2), 315–426 (2009)Google Scholar
  13. 13.
    Discacciati, M., Quarteroni, A., Miglio, E.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)CrossRefGoogle Scholar
  14. 14.
    Discacciati, M., Quarteroni, A., Valli, A.: Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45(3), 1246–1268 (2007). (electronic)CrossRefGoogle Scholar
  15. 15.
    Freund, J., Stenberg, R.: On weakly imposed boundary conditions for second order problems. In: Proceedings of the 9th International Conference Finite Elements in Fluids (1995)Google Scholar
  16. 16.
    Gatica, G.N., Oyarzua, R., Sayas, F.J.: Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comput. 80, 1911–1948 (2011)CrossRefGoogle Scholar
  17. 17.
    Jaeger, W., Mikelic, A.: On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)CrossRefGoogle Scholar
  18. 18.
    John, V., Matthies, G.: MooNMD—a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)CrossRefGoogle Scholar
  19. 19.
    Jones, I.: Low Reynolds number flow past a porous spherical shell. Math. Proc. Cambridge Philos. Soc 73, 231–238 (1973)CrossRefGoogle Scholar
  20. 20.
    Layton, W., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40, 2195–2218 (2003)CrossRefGoogle Scholar
  21. 21.
    Levy, T., Sanchez-Palencia, E.: On the boundary condition for fluid flow in porous media. Int. J. Eng. Sci. 13, 923–940 (1975)CrossRefGoogle Scholar
  22. 22.
    Mardal, K., Tai, X.C.,Winther, R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40, 1605–1631 (2002)CrossRefGoogle Scholar
  23. 23.
    Nitsche, J.: Über ein Variationsprinzip zur Lo¨sung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971). Collection of articles dedicated to Lothar Collatz on his 60th birthdayCrossRefGoogle Scholar
  24. 24.
    Riviére, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42, 1955–1977 (2005)CrossRefGoogle Scholar
  25. 25.
    Saffman, P.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 50, 93–101 (1971)Google Scholar
  26. 26.
    Urquiza, J., N’Dri, D., Garon, A., Delfour, M.: Coupling Stokes and Darcy equations. Appl. Numer. Mathe 58(5), 525–538 (2008)CrossRefGoogle Scholar
  27. 27.
    Xie, X., Xu, J., Xue, G.: Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. J. Comput. Math. 26, 437–455 (2008)Google Scholar
  28. 28.
    Zunino, P., D’Angelo, C.: Robust numerical approximation of coupled Stokes’ and Darcy’s flows applied to vascular hemodynamics and biochemical transport. ESAIM Math. Model. Numer. Anal. 45(3), 447–476 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alfonso Caiazzo
    • 1
  • Volker John
    • 1
    • 2
  • Ulrich Wilbrandt
    • 1
  1. 1.Weierstrass Institute for Applied Analysis and Stochastics (WIAS)BerlinGermany
  2. 2.Department of Mathematics and Computer ScienceFree University of BerlinBerlinGermany

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