Skip to main content
SpringerLink
Log in

Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem

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

Abstract

In this paper, we are concerned with space-time a posteriori error estimators for fully discrete solutions of linear parabolic problems. The mixed formulation with Raviart–Thomas finite element spaces is considered. A new second-order method in time is proposed so that mixed finite element spaces are permitted to change at different time levels. The new method can be viewed as a variant Crank–Nicolson (CN) scheme. Introducing a CN reconstruction appropriate for the mixed setting, we construct an a posteriori error estimator of second order in time for the variant CN mixed scheme. Various numerical examples are given to test our space-time adaptive algorithm and validate the theory proved in the paper. In addition, numerical results for backward Euler and CN schemes are presented to compare their performance in the time adaptivity setting over uniform/adaptive spatial meshes.

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

References

  1. Akrivis, G., Makridakis, C., Nochetto, R.: A posteriori error estimates for the CN method for parabolic equations. Math. Comput. 75(254), 511–531 (2006)

    Article  MATH  Google Scholar 

  2. Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aziz, A.K., Monk, P.: Continuous finite elements in space and time for the heat equation. Math. Comput. 52, 255–274 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Rheinboldt, W.C.: Aposteriori error estimates for the finite element method. Numer. Math. 12, 1597–1615 (1978)

    Google Scholar 

  5. Bansch, E., Karakatsani, F., Makridakis, Ch.: A posteriori error control for fully discrete CN schemes. SIAM J. Numer. Anal. 50(6), 2845–2872 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33(6), 2431–2444 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66(218), 465–476 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carstensen, C., Kim, D., Park, E.-J.: A priori and a posteriori pseudostress–velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49, 2501–2523 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cascon, J.M., Ferragut, L., Asensio, M.I.: Space-time adaptive algorithm for the mixed parabolic problem. Numer. Math. 103(3), 367–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clement, P.: Approximation by finite element functions using local regularization. RAIRO R–2, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  11. Douglas, J., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44(169), 39–52 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems IV: nonlinear problems. SIAM J. Numer. Anal. 2(6), 1729–1749 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1997)

    MATH  Google Scholar 

  15. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations, pp. 27–28. Springer, New York (1986)

    Book  MATH  Google Scholar 

  16. Johnson, C.: Numerical Solution of Partial Differential Equationns by the Finite Element Method. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  17. Johnson, C., Thomée, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 15, 41–78 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, D., Park, E.-J.: A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 197(6–8), 806–820 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, D., Park, E.-J.: Adaptive CN methods with dynamic finite-element spaces for parabolic problems. Discrete Contin. Dyn. Syst. Ser. B 10(4), 873–886 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kim, D., Park, E.-J.: A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations. SIAM J. Numer. Anal. 48(3), 1186–1207 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim, M.-Y., Milner, F.A., Park, E.-J.: Some observations on mixed methods for fully nonlinear parabolic problems in divergence form. Appl. Math. Lett. 9, 75–81 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, M.-Y., Park, E.-J., Thomas, S.G., Wheeler, M.F.: A multiscale mortar mixed finite element method for slightly compressible flows in porous media. J. Korean Math. Soc. 44(5), 1103–1119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Larson, M.G., Malqvist, A.: A posteriori error estimates for mixed finite element approximations of parabolic problems. Numer. Math. 118, 33–48 (2011)

    Article  MathSciNet  Google Scholar 

  24. Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41(4), 1585–1594 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Memon, S., Nataraj, N., Pani, A.K.: An a posteriori error analysis of mixed finite element Galerkin approximations to second order linear parabolic problems. SIAM J. Numer. Anal. 50(3), 1367–1393 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Milner, F.A., Park, E.-J.: A mixed finite element method for a strongly nonlinear second-order elliptic problem. Math. Comput. 64, 973–988 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nochetto, R.H., Schmidt, A., Verdi, C.: A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69, 1–24 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Østerby, O.: Five ways of reducing the CN oscillations. BIT Numer. Math. 43, 811–822 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Park, E.-J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21(2), 213–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Park, E.-J.: Mixed finite element methods for nonlinear second-order elliptic problems. SIAM J. Numer. Anal. 32(3), 865–885 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Raviart, P.A., Thomas, J.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of the Finite Elements Method. Lectures Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)

  32. Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43, 1021–1042 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wohlmuth, B.I., Hoppe, R.H.W.: A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart–Thomas elements. Math. Comput. 68, 1347–1378 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, D.Q.: Mixed methods with dynamic finite-element spaces for miscible displacement in porous media. J. Comput. Appl. Math. 30(3), 313–328 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which improved the quality of the paper.

Author information

Authors and Affiliations

  1. University College, Yonsei University, Seoul, 03722, Korea

    Dongho Kim

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 03722, Korea

    Eun-Jae Park

  3. Department of Mathematics, Yonsei University, Seoul, 03722, Korea

    Boyoon Seo

Authors
  1. Dongho Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eun-Jae Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boyoon Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eun-Jae Park.

Additional information

Dongho Kim was supported by NRF-2013R1A1A2007462. Eun-Jae Park was supported by NRF-2015R1A5A1009350 and NRF-2016R1A2B4014358.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, D., Park, EJ. & Seo, B. Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem. J Sci Comput 74, 1725–1756 (2018). https://doi.org/10.1007/s10915-017-0514-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0514-8

Keywords

Mathematics Subject Classification

Search

Navigation