Finite Element Method for the Stokes–Darcy Problem with a New Boundary Condition


This paper considers numerical methods for approximating and simulating the Stokes–Darcy problem, with a new boundary condition. We study a robust stabilized fully mixed discretization technique. This method ensures stability of the finite element scheme and does not use any Lagrange multipliers to introduce a stabilization term in the temporal Stokes–Darcy problem discretization. A correct finite element scheme is obtained and its convergence analysis is done. Finally, the efficiency and accuracy of these numerical methods are illustrated by different numerical tests.

This is a preview of subscription content, log in to check access.


  1. 1

    Adams, R.A. and Fournier, J.J.F., Sobolev Spaces, 2nd ed., Amsterdam: Elsevier, 2003.

  2. 2

    Ainsworth, M. and Oden, J., A Posteriori Error Estimators for the Stokes and Oseen Equations, SIAM J. Num. An., 1997, vol. 34, pp. 228–245.

  3. 3

    Arbogast, T. and Brunson, D.S., A Computational Method for Approximating a Darcy–Stokes System Governing a Vuggy Porous Medium,Comput. Geosci., 2007, vol. 11, pp. 207–218.

  4. 4

    Arnold, D.N., Brezzi, F., and Fortin, M., A Stable Finite Element for the Stokes Equations, Calcolo, 1984, vol. 21, pp. 337–344.

  5. 5

    Amaziane, B., El Ossmani, M., and Serres, C., Numerical Modeling of the Flow and Transport of Radionuclides in Heterogeneous Porous Media, Comput. Geosci., 2008, vol. 12, pp. 437–449.

  6. 6

    Babuska, I., Error-Bounds for Finite Element Method,Numer. Math., 1971, vol. 16, pp. 322–333.

  7. 7

    Badea, L., Discacciati, M., and Quarteroni, A., Numerical Analysis of the Navier–Stokes/Darcy Coupling, Numerische Math., 2010, vol. 115, pp. 195–227.

  8. 8

    Bank, R.E. and Welfert, B., A Posteriori Error Estimates for the Stokes Problem, SIAM J. Num. An., 1991, vol. 28, pp. 591–623.

  9. 9

    Benzi, M., Golub, G.H., and Liesen, J., Numerical Solution of Saddle Point Problems, Acta Numerica, 2005, vol. 14, pp. 1–137.

  10. 10

    Boubendir, Y. and Tlupova, S., Domain Decomposition Methods for Solving Stokes–Darcy Problems with Boundary Integrals, SIAM J. Sci. Comput., 2013, vol. 35, pp. B82–B106.

  11. 11

    Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, New York: Springer, 1991.

  12. 12

    Brezzi, F., Douglas, J., Jr., Fortin, M., and Marini, L.D., Efficient Rectangular Mixed Finite Elements in Two and Three Space Variables, Math. Model. Num. An., 1987, vol. 21, pp. 581–604.

  13. 13

    Brezzi, F., On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers,RAIRO: Num. An., 1974, vol. 8, pp. 129–151.

  14. 14

    Burman, E. and Hansbo, P., A Unified Stabilized Method for Stokes’ and Darcy’s Equations, J. Comput. Appl. Math., 2007, vol. 198, pp. 35–51.

  15. 15

    Cai, M., Mu, M., and Xu, J., Numerical Solution to a Mixed Navier–Stokes/Darcy Model by the Two-Grid Approach, SIAM J. Num. An., 2009, vol. 47, pp. 3325–3338.

  16. 16

    Camaño, J., Gatica, G.N., Oyarzúa, R., Ruiz-Baier, R., and Venegas, P., New Fully-Mixed Finite Element Methods for the Stokes–Darcy Coupling, Comput. Meth. Appl. Mech. Engin., 2015, vol. 295, pp. 362–395.

  17. 17

    Cao, Y., Gunzburger, M., He, X.M., and Wang, X., Robin-Robin Domain Decomposition Methods for the Steady-State Stokes–Darcy System with the Beavers–Joseph Interface Condition, Numerische Math., 2011, vol. 117, pp. 601–629.

  18. 18

    Cao, Y., Gunzburger, M., Hua, F., and Wang, X., Coupled Stokes–Darcy Model with Beavers–Joseph Interface Boundary Condition,Commun. Math. Sci., 2010, vol. 8, pp. 1–25.

  19. 19

    Carstensen, C. and Funken, S.A., A Posteriori Error Control in Low-Order Finite Element Discretizations of Incompressible Stationary Flow Problems, Math. Comput., 2001, vol. 70, pp. 1353–1381.

  20. 20

    Chavent, G. and Jaffre, J., Mathematical Models and Finite Elements in Reservoir Simulation, Netherlands: Elsevier, 1986.

  21. 21

    Discacciati, M., Miglio, E., and Quarteroni, A., Mathematical and Numerical Models for Coupling Surface and Groundwater Flows,Appl. Num. Math., 2002, vol. 43, pp. 57–74.

  22. 22

    Discacciati, M. and Quarteroni, A., Convergence Analysis of a Subdomain Iterative Method for the Finite Element Approximation of the Coupling of Stokes and Darcy Equations, Comput. Visualiz. Sci., 2004, vol. 6, pp. 93–103.

  23. 23

    Du, G. and Zuo, L., Local and Parallel Finite Element Method for the Mixed Navier–Stokes/Darcy Model with Beavers–Joseph Interface Conditions, Acta Mathematica Scientia, 2017, vol. 37B, pp. 1331–1347.

  24. 24

    Elakkad, A., Elkhalfi, A., and Guessous, N., An a Posteriori Error Estimate for Mixed Finite Element Approximations of the Navier–Stokes Equations, J. Korean Math. Soc., 2011, vol. 48, no. 3, pp. 529–550.

  25. 25

    Elman, H., Silvester, D., and Wathen, A., Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford: Oxford Univ. Press, 2014.

  26. 26

    Gatica, G.N., Oyarzúa, R., and Sayas, F.J., Analysis of Fully-Mixed Finite Element Methods for the Stokes–Darcy Coupled Problem, Math. Comput., 2011, vol. 80, pp. 1911–1948.

  27. 27

    Gatica, G.N., Oyarzúa, R., and Sayas, F.J., Convergence of a Family of Galerkin Discretizations for the Stokes–Darcy Coupled Problem, Num. Meth. Part. Diff. Eqs., 2011, vol. 27, pp. 721–748.

  28. 28

    Ghia, U., Ghia, K., and Shin, C., High-Re Solutions for Incompressible Flow Using the Navier–Stokes Equations and a Multigrid Method, J. Comput. Phys., 1982, vol. 48, pp. 387–395.

  29. 29

    Girault, V. and Rivière, B., DG Approximation of Coupled Navier–Stokes and Darcy Equations by Beaver–Joseph–Saffman Interface Condition, SIAM J. Num. An., 2009, vol. 47, pp. 2052–2089.

  30. 30

    He, X.M., Li, J., Lin, Y.P., and Ming, J., A Domain Decomposition Method for the Steady-State Navier–Stokes–Darcy Model with the Beavers–Joseph Interface Condition, SIAM J. Sci. Comput., 2015, vol. 37, pp. S264–S290.

  31. 31

    Hecht, F., Pironneau, O., Le Hyaric, A., and Ohtsuka, K., Freefem++, URL:

  32. 32

    Incompressible Computational Fluid Dynamics, Gunzburger, M. and Nicolaides, R., Eds., Cambridge: Cambridge Univ. Press, 1993.

  33. 33

    Lipnikov, K., Vassilev, D., and Yotov, I., Discontinuous Galerkin and Mimetic Finite Difference Methods for Coupled Stokes–Darcy Flows on Polygonal and Polyhedral Grids,Numerische Math., 2014, vol. 126, pp. 321–360.

  34. 34

    Mu, M. and Xu, J., A Two-Grid Method of a Mixed Stokes–Darcy Model for Coupling Fluid Flow with Porous Media Flow, SIAM J. Num. An., 2007, vol. 45, pp. 1801–1813.

  35. 35

    Mu, M. and Zhu, X., Decoupled Schemes for a Non-Stationary Mixed Stokes–Darcy Model, Math. Comput., 2010, vol. 79, pp. 707–731.

  36. 36

    Payne, L.E. and Straughan, B., Analysis of the Boundary Condition at the Interface between a Viscous Fluid and a Porous Medium and Related Modelling Questions, J. Math. Pures Appl., 1998, vol. 77, pp. 317–354.

  37. 37

    Pearson, J.W., Pestana, J., and Silvester, D.J., Refined Saddle-Point Preconditioners for Discretized Stokes Problems,Numerische Math., 2018, vol. 138, pp. 331–363; DOI: 10.1007/s00211-017-0908-4.

  38. 38

    Roberts, J. and Thomas, J.M., in Mixed and Hybrid Methods, Handbook of Numerical Analysis, Ciarlet, P. and Lions, J., Eds., vol. II: Finite Element Methods (part I), North Holland, 1990, pp. 523–639.

  39. 39

    Rui, H. and Zhang, R., A Unified Stabilized Mixed Finite Element Method for Coupling Stokes and Darcy Flows, Comput. Meth. Appl. Mech. Engin., 2009, vol. 198, pp. 2692–2699.

  40. 40

    Saffman, P., On the Boundary Condition at the Surface of a Porous Medium, Stud. Appl. Math., 1971, vol. 50, pp. 93–101.

  41. 41

    Shan, L. and Zheng, H., Partitioned Time Stepping Method for Fully Evolutionary Stokes–Darcy Flow with Beavers–Joseph Interface Conditions, SIAM J. Num. An., 2013, vol. 51, pp. 813–839.

  42. 42

    Urquiza, J.M., N’Dri, D., Garon, A., and Delfour, M.C., Coupling Stokes and Darcy Equations, Appl. Num. Math., 2008, vol. 58, pp. 525–538.

  43. 43

    Zuo, L. and Hou, Y., A Decoupling Two-Grid Algorithm for the Mixed Stokes–Darcy Model with the Beavers–Joseph Interface Condition,Num. Meth. Part. Diff. Eqs., 2014, vol. 30, pp. 1066–1082.

  44. 44

    Zuo, L. and Hou, Y., A Two-Grid Decoupling Method for the Mixed Stokes–Darcy Model, J. Comput. Appl. Math., 2015, vol. 275, pp. 139–147.

  45. 45

    Zuo, L. and Hou, Y., Numerical Analysis for the Mixed Navier–Stokes and Darcy Problem with the Beavers–Joseph Interface Condition, Num. Meth. Part. Diff. Eqs., 2015, vol. 31, pp. 1009–1030.

Download references

Author information



Corresponding authors

Correspondence to O. El Moutea or A. El Akkad.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

El Moutea, O., El Amri, H. & El Akkad, A. Finite Element Method for the Stokes–Darcy Problem with a New Boundary Condition. Numer. Analys. Appl. 13, 136–151 (2020).

Download citation


  • Stokes–Darcy problem
  • mixed finite element method
  • free flow
  • porous media flow
  • stabilized scheme