Skip to main content
SpringerLink
Log in

Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we analyze the supercloseness property of the streamline diffusion finite element method (SDFEM) on Shishkin triangular meshes, which is different from one in the case of rectangular meshes. The analysis depends on integral inequalities for the parts related to the diffusion in the bilinear form. Moreover, our result allows the construction of a simple postprocessing that yields a more accurate solution. Finally, numerical experiments support these theoretical results.

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

Similar content being viewed by others

References

  1. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, part I: Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14(1), 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, H., Lin, Q., Zhou, J., Wang, H.: Uniform error estimates for triangular finite element solutions of advection-diffusion equations. Adv Comput. Math. 38(1), 83–100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Franz, S., Kellogg, R.B., Stynes, M.: Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities. Math. Comp. 81(278), 661–685 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Franz, S., Linß, T., Roos, H.-G.: Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers. Appl. Numer. Math. 58(12), 1818–1829 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, W., Stynes, M.: Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection–diffusion problem. IMA J. Numer Anal. 17(1), 29–59 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (ed.) Finite Element Methods for Convection Dominated Flows, volume AMD 34, pp. 19–35. Amer. Soc. Mech. Engrs (ASME)., New York (1979)

  8. Lin, Q., Yan, N., Zhou, A.: A Rectangle Test for Interpolated Finite Elements. In: Proceedings Systems Science and Engineering (Hong Kong, 1991), pp. 217–229. Great Wall Culture Publishing, Whittier, CA (1991)

  9. Linß, T., Stynes, M.: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J. Math. Anal. Appl. 261(2), 604–632 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Linß, T., Stynes, M.: Numerical methods on Shishkin meshes for linear convection–diffusion problems. Comput. Methods Appl. Mech. Engrg. 190(28), 3527–3542 (2001)

    Article  MATH  Google Scholar 

  11. Roos, H.-G.: Superconvergence on a hybrid mesh for singularly perturbed problems with exponential layers. ZAMM Z. Angew. Math. Mech. 86(8), 649–655 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations Volume 24 of Springer Series in Computational Mathematics, second. Springer, Berlin (2008)

    MATH  Google Scholar 

  13. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shishkin, G.I.: Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations (In Russian). Second doctoral thesis. Keldysh Institute, Moscow (1990)

    Google Scholar 

  15. Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005)

  16. Stynes, M., O’Riordan, E.: A uniformly convergent G,alerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal Appl. 214(1), 36–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Stynes, M., Tobiska, L.: The SDFEM for a convection–diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41(5), 1620–1642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Stynes, M., Tobiska, L.: Using rectangular Q p elements in the SDFEM for a convection–diffusion problem with a boundary layer. Appl. Numer. Math. 58(12), 1789–1802 (2008)

  19. Zarin, H., Roos, H.-G.: Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers. Numer. Math. 100(4), 735–759 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, J., Liu, X.: Analysis of the SDFEM in a modified streamline diffusion norm for singularly perturbed convection diffusion problems. arXiv:1603.02099 (2016)

  21. Zhang, J., Mei, L., Chen, Y.: Pointwise estimates of the SDFEM for convection–diffusion problems with characteristic layers. Appl. Numer. Math. 64, 19–34 (2013)

  22. Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-D convection–diffusion problems. Math. Comp. 72(243), 1147–1177 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu, H., Zhang, Z.: Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer. Math. Comp. 83(286), 635–663 (2014)

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    Jin Zhang

  2. College of Science, Qilu University of Technology, Jinan, 250353, China

    Xiaowei Liu

Authors
  1. Jin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaowei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Zhang.

Additional information

Communicated by: Jinchao Xu

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Liu, X. Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers. Adv Comput Math 43, 759–775 (2017). https://doi.org/10.1007/s10444-016-9505-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-016-9505-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation