Skip to main content
SpringerLink
Log in

A posteriori error estimates for nonconforming streamline-diffusion finite element methods for convection-diffusion problems

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

We consider residual-based a posteriori error estimation for lowest-order nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. The resulting error estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. The error analysis is also shown to be applied to nonconforming finite element methods with face penalty and subgrid viscosity. Numerical results show that the estimator can be used to construct adaptive 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

Similar content being viewed by others

References

  1. Achchab, B., Fatini, M.E., Ern, A., Souissi, A.: A posteriori error estimates for subgrid viscosity stabilized approximations of convection-diffusion equations. Appl. Math. Lett. 22(9), 1418–1424 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Achdou, Y., Bernardi, C., Coquel, F.: A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96(1), 17–42 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Araya, R., Behrens, E., Rodríguez, R.: An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation. Appl. Numer. Math. 54(3–4), 491–503 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Araya, R., Poza, A.H., Stephan, E.P.: A hierarchical a posteriori error estimate for an advection-diffusion-reaction problem. Math. Mod. Meth. Appl. S. 15(7), 1119–1139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012–2033 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burman, E., Ern, A.: Continuous interior penalty \(hp\)-finite element methods for advection and advection-diffusion equations. Math. Comput. 76(259), 1119–1140 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ciarlet, P.: The finite element method for elliptic problems. North Holland, Amsterdam (1978)

    MATH  Google Scholar 

  8. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)

    MATH  Google Scholar 

  9. Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Modél. Math. Anal. Numér. 3, 33–75 (1973)

    MathSciNet  Google Scholar 

  10. El Alaoui, L., Ern, A.: Nonconforming finite element methods with subgrid viscosity applied to advection-diffusion-reaction equations. Numer. Meth. Part. D. E. 22(5), 1106–1126 (2006)

    Article  MATH  Google Scholar 

  11. El Alaoui, L., Ern, A., Burman, E.: A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations. IMA J. Numer. Anal. 27(1), 151–171 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hecht, F., Pironneau, O., Morice, J., Le Hyaric, A., Ohtsuka, K.: Freefem++ documentation, version 3.19-1. http://www.freefem.org/ff++ (2012)

  13. John, V., Matthies, G., Schieweck, F., Tobiska, L.: A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems. Comput. Method. Appl. M. 166(1), 85–97 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. John, V., Maubach, J., Tobiska, L.: Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78(2), 165–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Karakashian, O., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Knobloch, P., Tobiska, L.: The P1 mod element: a new nonconforming finite element for convection-diffusion problems. SIAM J. Numer. Anal. 41(2), 436–456 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kunert, G.: A posteriori error estimation for convection dominated problems on anisotropic meshes. Math. Method. Appl. Sci. 26(7), 589–617 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Matthies, G., Tobiska, L.: The streamline-diffusion method for conforming and nonconforming finite elements of lowest order applied to convection-diffusion problems. Computing 66(4), 343–364 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707–719 (1989)

    Article  MathSciNet  Google Scholar 

  20. Risch, U.: Superconvergence of a nonconforming low order finite element. Appl. Numer. Math. 54(3–4), 324–338 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sangalli, G.: A uniform analysis of nonsymmetric and coercive linear operators. SIAM J. Math. Anal. 36(6), 2033–2048 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Sangalli, G.: Robust a-posteriori estimator for advection-diffusion-reaction problems. Math. Comput. 77(261), 41–70 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stynes, M., Tobiska, L.: The streamline-diffusion method for nonconforming \({Q}_1^{rot}\) elements on rectangular tensor-product meshes. IMA J. Numer. Anal. 21(1), 123–142 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tobiska, L., Verfürth, R.: Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33(1), 107–127 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. arXiv:1402.5892 [math.NA] (2014)

  26. Verfürth, R.: A posteriori error estimators for convection-diffusion equations. Numer. Math. 80(4), 641–663 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Verfürth, R.: Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Verfürth, R.: A posteriori error estimation techniques for finite element methods. Oxford University Press, UK (2013)

    Book  MATH  Google Scholar 

  29. Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45(4), 1570–1599 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Vohralík, M.: Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods. Numer. Math. 111(1), 121–158 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66(217), 31–44 (1997)

    Article  MATH  Google Scholar 

  32. Zhou, G., Rannacher, R.: Pointwise superconvergence of the streamline diffusion finite-element method. Numer. Meth. Part. D. E. 12(1), 123–145 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referees for their suggestions on this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou , 450001, China

    Jikun Zhao, Shaochun Chen & Bei Zhang

Authors
  1. Jikun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaochun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jikun Zhao.

Additional information

This work is supported by National Natural Science Foundation of China (11371331).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Chen, S. & Zhang, B. A posteriori error estimates for nonconforming streamline-diffusion finite element methods for convection-diffusion problems. Calcolo 52, 407–424 (2015). https://doi.org/10.1007/s10092-014-0122-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-014-0122-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation