Stokes–Darcy Equations

  • Ulrich Wilbrandt
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


Let \(\varOmega \subset \mathbb {R}^d\) be a Lipschitz domain split into two disjoint nonempty subdomains Ωp and Ωf which are Lipschitz, too. The index p refers to the Darcy subdomain where a porous medium is modeled, while the index f refers to the free flow domain with a Stokes model.


  1. [BF91]
    Franco Brezzi and Michel Fortin. Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
  2. [Bra07b]
    Dietrich Braess. Finite elements. Cambridge University Press, Cambridge, third edition, 2007. Theory, fast solvers, and applications in elasticity theory, Translated from the German by Larry L. Schumaker.Google Scholar
  3. [BS08]
    Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008.Google Scholar
  4. [CGHW10]
    Yanzhao Cao, Max Gunzburger, Fei Hua, and Xiaoming Wang. Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci., 8(1):1–25, 2010.MathSciNetCrossRefGoogle Scholar
  5. [CGHW11]
    Wenbin Chen, Max Gunzburger, Fei Hua, and Xiaoming Wang. A parallel robin-robin domain decomposition method for the Stokes-Darcy system. SIAM J. Numerical Analysis, 49(3):1064–1084, 2011.MathSciNetCrossRefGoogle Scholar
  6. [CGHW14]
    Yanzhao Cao, Max Gunzburger, Xiaoming He, and Xiaoming Wang. Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems. Math. Comp., 83(288):1617–1644, 2014.MathSciNetCrossRefGoogle Scholar
  7. [Cia02]
    Philippe G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].Google Scholar
  8. [CJW14]
    Alfonso Caiazzo, Volker John, and Ulrich Wilbrandt. On classical iterative subdomain methods for the Stokes-Darcy problem. Comput. Geosci., 18(5):711–728, 2014.MathSciNetCrossRefGoogle Scholar
  9. [DQV07]
    Marco Discacciati, Alfio Quarteroni, and Alberto Valli. Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal., 45(3):1246–1268, 2007.MathSciNetCrossRefGoogle Scholar
  10. [DZ11]
    Carlo D’Angelo and Paolo Zunino. 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.MathSciNetCrossRefGoogle Scholar
  11. [GOS11a]
    Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas. Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comp., 80(276):1911–1948, 2011.MathSciNetCrossRefGoogle Scholar
  12. [GOS11b]
    Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas. Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem. Numer. Methods Partial Differential Equations, 27(3):721–748, 2011.MathSciNetCrossRefGoogle Scholar
  13. [GS07]
    Juan Galvis and Marcus Sarkis. Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal., 26:350–384, 2007.MathSciNetzbMATHGoogle Scholar
  14. [JM00]
    Willi Jäger and Andro Mikelić. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math., 60(4):1111–1127, 2000.MathSciNetCrossRefGoogle Scholar
  15. [Jon73]
    I.P. Jones. Low Reynolds number flow past a porous spherical shell. Math. Proc. Cambridge Philos. Soc., 73:231–238, 1973.CrossRefGoogle Scholar
  16. [LSY02]
    William J. Layton, Friedhelm Schieweck, and Ivan Yotov. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal., 40(6):2195–2218 (2003), 2002.MathSciNetCrossRefGoogle Scholar
  17. [MMS15]
    Antonio Márquez, Salim Meddahi, and Francisco-Javier Sayas. Strong coupling of finite element methods for the Stokes-Darcy problem. IMA J. Numer. Anal., 35(2):969–988, 2015.MathSciNetCrossRefGoogle Scholar
  18. [RM14]
    Iryna Rybak and Jim Magiera. A multiple-time-step technique for coupled free flow and porous medium systems. J. Comput. Phys., 272:327–342, 2014.MathSciNetCrossRefGoogle Scholar
  19. [Saf71]
    P.G. Saffman. On the boundary condition at the interface of a porous medium. Stud. Appl. Math., 50:93–101, 1971.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ulrich Wilbrandt
    • 1
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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