Skip to main content
SpringerLink
Log in

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we consider a decoupled and mass-conservative numerical scheme for the incompressible miscible displacement problem in porous media by employing multipoint flux mixed finite element method for velocity-pressure equation and mass-conservation characteristic finite element method for concentration equation. This feature is designed to avoid handling saddle-point system and keep mass globally. Convergence and L2 error estimates of full discrete scheme are obtained. Numerical experiments are also presented to verify the correctness of 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

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Ewing, R., Russell, T., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Engrg. 47, 73–92 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Russell, T.: Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal. 22, 970–1013 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ewing, R., Wheeler, M.F.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17, 351–365 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Douglas, Jr., Ewing, R., Wheeler, M.F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numer. 17, 17–33 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Douglas, Jr., Ewing, R., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer. 17, 249–265 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, H., Liang, D., Ewing, R.E.: An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comput. 22, 561–581 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, J., Riviere, B.: Numerical solutions of the incompressible miscible displacement equations in heterogeneous media. Comput. Methods Appl. Mech. Eng. 292, 107–121 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Xu, W., Liang, D., Rui, H.: A multipoint flux mixed finite element method for the compressible Darcy-Forchheimer models. Appl. Math. Comput. 315, 259–277 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Chen, Y., Hu, H.: Two-grid method for miscible displacement problem by mixed finite element methods and mixed finite element method of characteristics. Commun. Comput. Phys. 19, 1503–1528 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, H.: Two-grid method for compressible miscible displacement problem by mixed finite element methods and expanded mixed finite element method of characteristics. Numer. Algorithm. 89, 611–636 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee, S., Wheeler, M.: Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization. J. Comput. Phys. 331, 19–37 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brezzi, F., Douglas, Jr., Marini, L.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 88, 217–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods Springer Series in Computational Mathematics, vol. 15. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  14. Wheeler, M., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44, 2082–2106 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu, W.: Cell-centered finite difference method for parabolic equation. Appl. Math. Comput. 235, 66–79 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Xu, W., Liang, D., Rui, H., Li, X.: A multipoint flux mixed finite element method for Darcy-Forchheimer incompressible miscible displacement problem. J. Sci. Comput. 82(1), 1–20 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xu, W., Liang, D., Rui, H.: A multipoint flux mixed finite element method for the compressible Darcy-Forchheimer models. Appl. Math. Comput. 315, 259–277 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Douglas, Jr., Russell, T.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristic with finite element of finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rui, H.: A conservative characteristic finite volume element method for solution of the advection-diffusion equation. Comput. Methods Appl. Mech. Engrg. 197(45-48), 3862–3869 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rui, H., Tabata, M.: A mass-conservative finite element scheme for convection-diffusion problems. J. Sci. Comput. 43, 416–432 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, X., Rui, H.: A MCC finite element approximation of incompressible miscible displacement in porous media. Comput. Math. Appl. 70, 750–764 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, X., Rui, H., Xu, W.: A new MCC-MFE method for compressible miscible displacement in porous media. J. Comput. Appl. Math. 302, 139–156 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11801293), the Natural Science Foundation of Shandong Province (ZR2020MA049), and the Education and Industry Integration Pilot Project Basic Research Project of Qilu University of Technology (Shandong Academy of Sciences) (2022PY058).

Author information

Author notes

    Authors and Affiliations

    1. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

      Xindong Li, Mingyang Du & Wenwen Xu

    Authors
    1. Xindong Li
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Mingyang Du
      View author publications

      You can also search for this author in PubMed Google Scholar

    3. Wenwen Xu
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Xindong Li.

    Ethics declarations

    Conflict of interest

    The authors declare no competing interests.

    Additional information

    Publisher’s note

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

    Xindong Li, Mingyang Du, and Wenwen Xu contributed equally to this work.

    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

    Li, X., Du, M. & Xu, W. A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem. Numer Algor 93, 1795–1810 (2023). https://doi.org/10.1007/s11075-022-01489-3

    Download citation

    • Received:

    • Accepted:

    • Published:

    • Issue Date:

    • DOI: https://doi.org/10.1007/s11075-022-01489-3

    Keywords

    Search

    Navigation