Journal of Scientific Computing

, Volume 74, Issue 2, pp 693–727 | Cite as

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

Article

Abstract

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.

Keywords

Non-stationary Stokes–Darcy Discontinuous finite volume element method Beavers–Joseph–Saffman–Jones condition Error estimates 

References

  1. 1.
    Arbogast, T., Gomez, M.: A discretization and multigrid solver for a Darcy–Stokes system of three dimensional vuggy porous media. Comput. Geosci. 13(3), 331–348 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arbogast, T., Lehr, H.L.: Homogenization of a Darcy–Stokes system modeling vuggy porous media. Comput. Geosci. 10(3), 291–302 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Babuška, I., Gatica, G.N.: A residual-based a posteriori error estimator for the Stokes–Darcy coupled problem. SIAM J. Numer. Anal. 48(2), 498–523 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Badia, S., Codina, R., Gudi, T., Guzmán, J.: Error analysis of discontinuous galerkin methods for the stokes problem under minimal regularity. IMA J. Numer. Anal. 34(2), 800–819 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bernardi, C., Hecht, F., Mghazli, Z.: A new finite element discretization of the stokes problem coupled with darcy equations. IMA J. Numer. Anal. 30, 61–93 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bernardi, C., Rebollo, T.Chacón, Hecht, F., Mghazli, Z., Mghazli, Z.: Mortar finite element discretization of a model coupling Darcy and Stokes equations. Math. Model. Numer. Anal. 42, 375–410 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bi, C., Geng, J.: Discontinuous finite volume element method for parabolic problems. Numer. Methods Partial Differ. Equ. 26(2), 367–383 (2010)MathSciNetMATHGoogle Scholar
  9. 9.
    Bi, C., Geng, J.: A discontinuous finite volume element method for second-order elliptic problems. Numer. Methods Partial Differ. Equ. 28(2), 425–440 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boubendir, Y., Tlupova, S.: Stokes–Darcy boundary integral solutions using preconditioners. J. Comput. Phys. 228(23), 8627–8641 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bürgera, R., Kumarb, S., Ruiz-Baier, R.: Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation. J. Comput. Phys. 299, 446–471 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cai, Z., Mandel, J., McCormick, S.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28(2), 392–402 (1991)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cai, Z., McCormick, S.: On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal. 27(3), 636–655 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Camano, J., Gatica, G.N., Oyarzua, R., Ruiz-Baier, R., Venegas, P.: New fully-mixed finite element methods for the Stokes–Darcy coupling. Comput. Methods Appl. Mech. Eng. 295, 362–395 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cao, Y., Gunzburger, M., He, X.-M., Wang, X.: Robin–Robin domain decomposition methods for the steady Stokes–Darcy model with Beaver–Joseph interface condition. Numer. Math. 117(4), 601–629 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cao, Y., Gunzburger, M., He, X.-M., Wang, X.: Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems. Math. Comput. 83(288), 1617–1644 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, X., Zhao, W.: Finite element approximation for Stokes–Darcy flow with Beavers–Joseph interface conditions. SIAM. J. Numer. Anal. 47(6), 4239–4256 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Carstensen, C., Nataraj, N., Pani, A.K.: Comparison results and unified analysis for first-order finite volume element methods for a poisson model problem. IMA J. Numer. Anal. 36(3), 1120–1142 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Çeşmelioğlu, A., Rivière, B.: Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J. Sci. Comput. 40(1–3), 115–140 (2009)MathSciNetMATHGoogle Scholar
  21. 21.
    Chatzipantelidis, P., Ginting, V., Lazarov, R.D.: A finite volume element method for a non-linear elliptic problem. Numer. Linear Algebra Appl. 12(5–6), 515–546 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Methods Partial Differ. Equ. 20(5), 650–674 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Chen, Z., Chen, H.: Pointwise error estimates of discontinuous galerkin methods with penalty for second-order elliptic problems. SIAM J. Numer. Anal. 42(3), 1146–1166 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chou, S.H., Ye, X.: Unified analysis of finite volume methods for second order elliptic problems. SIAM J. Numer. Anal. 45(4), 1639–1653 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Cui, M., Ye, X.: Unified analysis of finite volume methods for the Stokes equations. SIAM J. Numer. Anal. 48(3), 824–839 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43(1–2), 57–74 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ervin, V.J., Jenkins, E.W., Sun, S.: Coupling nonlinear Stokes and Darcy flow using mortar finite elements. Appl. Numer. Math. 61(11), 1198–1222 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes–Darcy equations. Electron. Trans. Numer. Anal. 26, 350–384 (2007)MathSciNetMATHGoogle Scholar
  31. 31.
    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(1), 86–108 (2009)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Gatica, G.N., Oyarzúa, R., Sayas, F.J.: A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes–Darcy coupled problem. Comput. Methods Appl. Mech. Eng. 200(21–22), 1877–1891 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Girault, V., Rivière, B.: DG approximation of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition. SIAM J. Numer. Anal. 47(3), 2052–2089 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Girault, V., Riviere, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74(249), 53–84 (2005)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Girault, V., Vassilev, D., Yotov, I.: Mortar multiscale finite element methods for Stokes–Darcy flows. Numer. Math. 127(1), 93–165 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Hansboa, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191(17–18), 1895–1908 (2002)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Hanspal, N., Waghode, A., Nassehi, V., Wakeman, R.: Numerical analysis of coupled Stokes/Darcy flow in industrial filtrations. Transp. Porous Media 64, 73–101 (2006)CrossRefMATHGoogle Scholar
  38. 38.
    He, X.-M., Li, J., Lin, Y., Ming, J.: A domain decomposition method for the steady-state Navier–Stokes–Darcy model with Beavers–Joseph interface condition. SIAM J. Sci. Comput. 37(5), S264–S290 (2015)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Hessari, P.: Pseudospectral least squares method for Stokes–Darcy equations. SIAM J. Numer. Anal. 53(3), 1195–1213 (2015)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Hoppe, R., Porta, P., Vassilevski, Y.: Computational issues related to iterative coupling of subsurface and channel flows. Calcolo 44(1), 1–20 (2007)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Huang, P., Chen, J., Cai, M.: A mixed and nonconforming fem with nonmatching meshes for a coupled Stokes–Darcy model. J. Sci. Comput. 53, 377–394 (2012)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kanschat, G., Riviére, B.: A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Comput. Phys. 229, 5933–5943 (2010)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Kumar, S.: A mixed and discontinuous galerkin finite volume element method for incompressible miscible displacement problems in porous media. Numer. Methods Partial Differ. Equ. 28(4), 1354–1381 (2012)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kumar, S., Nataraj, N., Pani, A.K.: Discontinuous finite volume element methods for second order linear elliptic problems. Numer. Methods Partial Differ. Equ. 25, 1402–1424 (2009)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Kumarb, S., Ruiz-Baier, R.: Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65(3), 956–978 (2015)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Layton, W., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows. SIAM J. Numer. Anal. 51(1), 248–272 (2013)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2002)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Li, R., Li, J., Chen, Z., Gao, Y.: A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem. J. Comput. Appl. Math. 292, 92–104 (2016)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Li, R., Li, J., He, X., Chen, Z.: A stabilized finite volume element method for a coupled Stokes–Darcy problem. Appl. Numer. Math. (submitted)Google Scholar
  50. 50.
    Lipnikov, K., Vassilev, D., Yotov, I.: Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes–Darcy flows on polygonal and polyhedral grids. Numer. Math. 126(2), 321–360 (2014)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Liu, J., Mu, L., Ye, X.: An adaptive discontinuous finite volume method for elliptic problems. J. Comput. Appl. Math. 235(18), 5422–5431 (2011)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Liu, J., Mu, L., Ye, X., Jari, R.: Convergence of the discontinuous finite volume method for elliptic problems with minimal regularity. J. Comput. Appl. Math. 236(17), 4537–4546 (2012)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Márquez, A., Meddahi, S., Sayas, F.J.: Strong coupling of finite element methods for the Stokes–Darcy problem. IMA J. Numer. Anal. 35(2), 969–988 (2015)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Mu, L., Jari, R.: A posteriori error analysis for discontinuous finite volume methods of elliptic interface problems. J. Comput. Appl. Math. 255, 529–543 (2014)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Mu, M., Xu, J.: A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45(5), 1801–1813 (2007)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Mu, M., Zhu, X.: Decoupled schemes for a non-stationary mixed Stokes–Darcy model. Math. Comput. 79(270), 707–731 (2010)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Münzenmaier, S., Starke, G.: First-order system least squares for coupled Stokes–Darcy flow. SIAM J. Numer. Anal. 49(1), 387–404 (2011)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Nassehi, V.: Modelling of combined Navier–Stokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 53, 1253–1265 (1998)CrossRefGoogle Scholar
  59. 59.
    Rivière, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22(23), 479–500 (2005)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Shan, L., Zheng, H.: Partitioned time stepping method for fully evolutionary Stokes–Darcy flow with Beavers–Joseph interface conditions. SIAM J. Numer. Anal. 51(2), 813–839 (2013)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Shan, L., Zheng, H., Layton, W.: A decoupling method with different sub-domain time steps for the nonstationary Stokes–Darcy model. Numer. Methods Partial Differ. Equ. 29(2), 549–583 (2013)CrossRefMATHGoogle Scholar
  62. 62.
    Tlupova, S., Cortez, R.: Boundary integral solutions of coupled Stokes and Darcy flows. J. Comput. Phys. 228(1), 158–179 (2009)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Vassilev, D., Wang, C., Yotov, I.: Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 268, 264–283 (2014)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Wang, G., He, Y., Li, R.: Discontinuous finite volume methods for the stationary Stokes–Darcy problem. Int. J. Numer. Meth. Eng. 107(5), 395–418 (2016)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Wang, J., Wang, Y., Ye, X.: A unified a posteriori error estimator for finite volume methods for the Stokes equations. Math. Methods Appl. Sci. doi: 10.1002/mma.2871
  66. 66.
    Ye, X.: A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal. 42(3), 1062–1072 (2004)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Ye, X.: A discontinuous finite volume method for the Stokes problems. SIAM J. Numer. Anal. 44(1), 183–198 (2006)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Yin, Z., Jiang, Z., Xu, Q.: A discontinuous finite volume method for the Darcy–Stokes equations. J. Appl. Math. 761242–761258, 2012 (2012)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics, School of Arts and SciencesShaanxi University of Science and TechnologyXi’anPeople’s Republic of China
  3. 3.Department of MathematicsBaoji University of Arts and SciencesBaojiPeople’s Republic of China
  4. 4.Department of Chemical & Petroleum Engineering, Schulich School of EngineeringUniversity of CalgaryAlbertaCanada

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