Journal of Scientific Computing

, Volume 77, Issue 3, pp 1605–1620 | Cite as

A Strongly Conservative Hybrid DG/Mixed FEM for the Coupling of Stokes and Darcy Flow

  • Guosheng FuEmail author
  • Christoph Lehrenfeld


We consider the coupling of free and porous media flow governed by Stokes and Darcy equations with the Beavers–Joseph–Saffman interface condition. This model is discretized using a divergence-conforming finite element for the velocities in the whole domain. Hybrid discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. The discretization achieves mass conservation in the sense of \(H(\mathrm {div},\Omega )\), and we obtain optimal velocity convergence. Numerical results are presented to validate the theoretical findings.


HDG Stokes–Darcy Divergence-conforming Beavers–Joseph–Saffman 

Mathematics Subject Classification

65N30 65N12 76S05 76D07 


  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 20, 197–207 (1967)CrossRefGoogle Scholar
  3. 3.
    Bernardi, C., Hecht, F., Pironneau, O.: Coupling Darcy and Stokes equations for porous media with cracks. M2AN Math. Model. Numer. Anal. 39, 7–35 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bernardi, C., Rebollo, T.C., Hecht, F., Mghazli, Z.: Mortar finite element discretization of a model coupling Darcy and Stokes equations. M2AN Math. Model. Numer. Anal. 42, 375–410 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Brenner, S.C.: Korn’s inequalities for piecewise H 1 vector fields. Math. Comput. 73, 1067–1087 (2004)CrossRefGoogle Scholar
  7. 7.
    Burman, E., Hansbo, P.: Stabilized Crouzeix–Raviart element for the Darcy-Stokes problem. Numer. Methods Partial Differ. Eqs. 21, 986–997 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burman, E., Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations. J. Comput. Appl. Math. 198, 35–51 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cockburn, B., Sayas, F.-J.: Divergence-conforming HDG methods for Stokes flow. Math. Comput. 83, 1571–1598 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74, 1067–1095 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. J. Sci. Comput. 31, 61–73 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Correa, M., Loula, A.: A unified mixed formulation naturally coupling Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 198, 2710–2722 (2009)CrossRefGoogle Scholar
  13. 13.
    Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002). 19th Dundee Biennial Conference on Numerical Analysis (2001)Google Scholar
  14. 14.
    Ervin, V.J., Jenkins, E.W., Sun, S.: Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47, 929–952 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fu, G., Jin, Y., Qiu, W.: Parameter-free superconvergent h(div)-conforming HDG methods for the brinkman equations. arXiv preprint arXiv:1607.07662 (2016)
  16. 16.
    Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes–Darcy equations. Electron. Trans. Numer. Anal. 26, 350–384 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gatica, G.N., Meddahi, S., Oyarzúa, R.: A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 29, 86–108 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gatica, G.N., Oyarzúa, R., Sayas, F.-J.: Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem. Numer. Methods Partial Differ. Eqs. 27, 721–748 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin (1986)CrossRefGoogle Scholar
  20. 20.
    Girault, V., Kanschat, G., Rivière, B.: Error analysis for a monolithic discretization of coupled Darcy and Stokes problems. J. Numer. Math. 22, 109–142 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kanschat, G., Rivière, B.: A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Comput. Phys. 229, 5933–5943 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(2002), 2195–2218 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lederer, P.L., Lehrenfeld, C., Schöberl, J.: Hybrid discontinuous Galerkin methods with relaxed h(div)-conformity for incompressible flows. Part I. arXiv preprint arXiv:1707.02782 (2017)
  24. 24.
    Lehrenfeld, C.: Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Diploma Thesis, MathCCES/IGPM, RWTH Aachen (2010)Google Scholar
  25. 25.
    Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nassehi, V.: Modelling of combined NavierStokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53, 1253–1265 (1998)CrossRefGoogle Scholar
  28. 28.
    Nassehi, V., Waghode, A.N., Hanspal, N.S., Wakeman, R.J.: Mathematical modelling of flow through pleated cartridge filters. In: Progress in industrial mathematics at ECMI 2004, vol. 8 of Mathematics in Industry, pp. 298–302. Springer, Berlin (2006)CrossRefGoogle Scholar
  29. 29.
    Oikawa, I.: A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65, 327–340 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87, 69–93 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22(23), 479–500 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42, 1959–1977 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rui, H., Zhang, R.: A unified stabilized mixed finite element method for coupling Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 198, 2692–2699 (2009)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 292–315 (1971)Google Scholar
  35. 35.
    Schöberl, J.: C++11 Implementation of Finite Elements in NGSolve, ASC Report 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014)Google Scholar
  36. 36.
    Schroeder, P.W., Linke, A., Lehrenfeld, C., Lube, G.: Towards computable flows and robust estimates for inf-sup stable fem applied to the time-dependent incompressible Navier–Stokes equations. arXiv preprint arXiv:1709.03063 (2017)
  37. 37.
    Urquiza, J.M., N’Dri, D., Garon, A., Delfour, M.C.: Coupling Stokes and Darcy equations. Appl. Numer. Math. 58, 525–538 (2008)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Warburton, T., Hesthaven, J.S.: On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192, 2765–2773 (2003)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Xie, X., Xu, J., Xue, G.: Uniformly-stable finite element methods for Darcy–Stokes–Brinkman models. J. Comput. Math. 26, 437–455 (2008)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Institut für Numerische und Angewandte MathematikGöttingenGermany

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