Skip to main content
SpringerLink
Log in

Optimal Convergence Analysis of a Fully Discrete Scheme for the Stochastic Stokes–Darcy Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we consider numerical analysis for a fully discrete scheme for the stochastic Stokes–Darcy equations with multiplicative noise. Implicit Euler scheme is used for the time discretization, and interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM\(_1\)–P\(_0\) finite element space is used for the space discretization. Physical interface conditions are imposed to couple the fluid equations in free fluid and porous media regions. It is proved that the implicit Euler scheme for the stochastic Stokes–Darcy equations is unconditionally stable. Under usual assumptions for the multiplicative noise and regularity of the velocity, we present the optimal convergence analysis in both time and space discretizations. Moreover, our stability result and error estimates for the velocity are independent of pressure. Numerical results are given to verify the theoretical analysis.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data Availability

All data generated or analyzed during this study are included in this manuscript.

References

  1. Ambartsumyan, I., Khattatov, E., Wang, C., Yotov, I.: Stochastic multiscale flux basis for Stokes–Darcy flows. J. Comput. Phys. 401, 109011 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arbogast, T., Brunson, D.S.: A computational method for approximating a Darcy–Stokes system governing a vuggy porous media. Comput. Geosci. 11, 207–218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arbogast, T., Lehr, H.L.: Homogenization of a Darcy–Stokes system modeling vuggy porous media. Comput. Geosci. 10, 291–302 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

    Article  Google Scholar 

  5. Breit, D., Dogson, A.: Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations. Numer. Math. 147, 553–578 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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, 4239–4256 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carelli, E., Hausenblas, E., Prohl, A.: Time-splitting methods to solve the stochastic incompressible Stokes equations. SIAM J. Numer. Anal. 50, 2917–2939 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carelli, E., Prohl, A.: Rates of convergence for discretizations of the stochastic incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 50, 2467–2496 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chandesris, M., Jamet, D.: Boundary conditions at a planar fluid porous interface for a Poiseuille flow. Int. J. Heat Mass Transf. 49, 2137–2150 (2006)

    Article  MATH  Google Scholar 

  11. Chen, H., Wang, X.-P.: A one-domain approach for modeling and simulation of free fluid over a porous media. J. Comput. Phys. 259, 650–671 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  13. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  14. Dixon, J., McKee, S.: Weakly singular Gronwall inequalities. ZAMM Z. Angew. Math. Mech. 66, 535–544 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ervin, V.J., Jenkins, E.W., Sun, S.: Coupled generalized nonlinear Stokes flow with flow through a porous media. SIAM J. Numer. Anal. 47, 929–952 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feng, X., Prohl, A., Vo, L.: Optimally convergent mixed finite element methods for the stochastic Stokes equations. IMA J. Numer. Anal. 41, 2280–2310 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng, X., Qiu, H.: Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. J. Sci. Comput. 88, 1–25 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feng, X., Vo, L.: Analysis of Chorin-type projection methods for the stochastic Stokes equations with general multiplicative noises. Stoch. PDE: Anal. Comp. (2022). https://doi.org/10.1007/s40072-021-00228-4

    Article  Google Scholar 

  19. Gao, Y., He, X., Mei, L., Yang, X.: Decoupled, linear, and energy stable finite element method for the Cahn–Hilliard–Navier–Stokes–Darcy phase field model. SIAM J. Sci. Comput. 40, B110–B137 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goyeau, B., Lhuillier, D., Gobin, D., Velarde, M.G.: Momentum transport at a fluid–porous interface. Int. J. Heat Mass Transf. 46, 4071–4081 (2003)

    Article  MATH  Google Scholar 

  21. Han, D., He, X., Wang, Q., Wu, Y.: Existence and weak-strong uniqueness of solutions to the Cahn–Hilliard–Navier–Stokes–Darcy system in superposed free flow and porous media. Nonlinear Anal. 211, 112411 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hanspal, N.S., Waghode, A.N., Nassehi, V., Wakeman, R.J.: Numerical analysis of coupled Stokes/Darcy flow in industrial filtrations. Transp. Porous Media 64, 73–101 (2006)

    Article  MATH  Google Scholar 

  23. Hausenblas, E., Randrianasolo, T.A.: Time-discretization of stochastic 2-D Navier–Stokes equations with a penalty-projection method. Numer. Math. 143, 339–378 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Karper, T., Mardal, K.-A., Winther, R.: Unified finite element discretizations of coupled Darcy–Stokes flow. Numer. Methods Partial Differ. Equ. 25, 311–326 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kruse, R.: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  28. Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  29. Kumar, P., Luo, P., Gaspar, F.J., Oosterlee, C.W.: A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system. J. Comput. Phys. 371, 382–408 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  31. Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs: Galerkin Approximation and Finite Elements. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  32. Márquez, A., Meddahi, S., Sayas, F.-J.: Strong coupling of finite element methods for the Stokes–Darcy problem. IMA J. Numer. Anal. 35, 969–988 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nield, D.A.: The Beavers-Joseph boundary condition and related matters: a historical and critical note. Trans. Porous Media 78, 537–540 (2009)

    Article  MathSciNet  Google Scholar 

  34. Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    MATH  Google Scholar 

  35. Roman, L.J., Sarkis, M.: Stochastic Galerkin method for elliptic SPDEs: a white noise approach. Discrete Contin. Dyn. Syst. Ser. B 6, 941–955 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Saffman, P.G.: On the boundary condition at the interface of a porous media. Stud. Appl. Math. 1, 93–101 (1971)

    Article  MATH  Google Scholar 

  37. Tartakovsky, A.M., Tartakovsky, D.M., Meakin, P.: Stochastic Langevin model for flow and transport in porous media. Phys. Rev. Lett. 101, 044502 (2008)

    Article  Google Scholar 

  38. Yang, Z., Li, X., He, X., Ming, J.: A stochastic collocation method based on sparse grids for a stochastic Stokes–Darcy model. Discrete Contin. Dyn. Syst. Ser. S 15, 893–912 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, Z., Ming, J., Qiu, C., Li, M., He, X.: A multigrid multilevel Monte Carlo method for Stokes–Darcy model with random hydraulic conductivity and Beavers–Joseph condition. J. Sci. Comput. 90, 68 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, London (2002)

    Google Scholar 

Download references

Funding

This work was supported by the NSF of China (Grant Nos. 12122115, 11771363, 12271457).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian, 361005, China

    Yahong Xiang, Can Huang & Huangxin Chen

Authors
  1. Yahong Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Can Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huangxin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huangxin Chen.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiang, Y., Huang, C. & Chen, H. Optimal Convergence Analysis of a Fully Discrete Scheme for the Stochastic Stokes–Darcy Equations. J Sci Comput 94, 13 (2023). https://doi.org/10.1007/s10915-022-02057-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-02057-6

Keywords

Mathematics Subject Classification

Search

Navigation