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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  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)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 20, 197–207 (1967)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Book  Google Scholar 

  6. 6.

    Brenner, S.C.: Korn’s inequalities for piecewise H 1 vector fields. Math. Comput. 73, 1067–1087 (2004)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cockburn, B., Sayas, F.-J.: Divergence-conforming HDG methods for Stokes flow. Math. Comput. 83, 1571–1598 (2014)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

  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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin (1986)

    Book  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

  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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Nassehi, V.: Modelling of combined NavierStokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53, 1253–1265 (1998)

    Article  Google 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)

    Chapter  Google Scholar 

  29. 29.

    Oikawa, I.: A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65, 327–340 (2015)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42, 1959–1977 (2005)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

  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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Guosheng Fu.

Additional information

Dedicated to the 60th birthday of Professor Bernardo Cockburn.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fu, G., Lehrenfeld, C. A Strongly Conservative Hybrid DG/Mixed FEM for the Coupling of Stokes and Darcy Flow. J Sci Comput 77, 1605–1620 (2018). https://doi.org/10.1007/s10915-018-0691-0

Download citation

Keywords

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

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 76S05
  • 76D07