Numerical Algorithms

, Volume 78, Issue 2, pp 465–483 | Cite as

Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with characteristic layers

  • Xiaowei Liu
  • Jin Zhang
Original Paper


In this paper, we present pointwise estimates of the streamline diffusion finite element method (SDFEM) for conforming piecewise linears on Shishkin triangular meshes. The method is applied to a model singularly perturbed convection-diffusion problem with characteristic layers. Using a new variant of artificial crosswind diffusion, we prove that uniformly pointwise error bounds away from the layers are of order almost 7/4 (up to a logarithmic factor). In some cases, the convergence order is almost 15/8. Our analysis depends on discrete Green’s functions and sharp estimates of the diffusion and convection parts in the bilinear form. Finally, numerical experiments support our theoretical results.


Convection–diffusion Characteristic layers Shishkin triangular mesh SDFEM Pointwise error 

Mathematics Subject Classification (2010)

65N12 65N30 65N50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank two unknown referees for some perceptive comments that led them to improve this paper.


  1. 1.
    Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47(3), 277–293 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)CrossRefGoogle Scholar
  4. 4.
    Donea, J., Huerta, A.: Finite Element Methods for Flow Problems Wiley Online Library. Wiley, New York (2003)CrossRefGoogle Scholar
  5. 5.
    Elman, H.C., Ramage, A.: An analysis of smoothing effects of upwinding strategies for the convection–diffusion equation. SIAM J. Numer Anal. 40(1), 254–281 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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. American Society of Mechanical Engineers (ASME), New York (1979)Google Scholar
  7. 7.
    Johnson, C., Schatz, A.H., Wahlbin, L.B.: Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comp. 49(179), 25–38 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kellogg, R.B., Stynes, M.: Corner singularities and boundary layers in a simple convection-diffusion problem. J. Differ. Equ. 213(1), 81–120 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kellogg, R.B., Stynes, M.: Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem. Appl. Math. Lett. 20(5), 539–544 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lin, Q., Yan, N., Zhou, A.: A Rectangle Test for Interpolated Finite Elements. In: Proceedings Systems Science and Systems Engineering (Hong Kong, 1991), pp 217–229 . Great Wall Culture Publishing, Whittier, CA (1991)Google Scholar
  11. 11.
    Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems Vol. 1985 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Linß, T., Stynes, M.: Numerical methods on Shishkin meshes for linear convection–diffusion problems. Comput. Methods Appl. Mech. Engrg. 190(28), 3527–3542 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Linß, T., Stynes, M.: The SDFEM on Shishkin meshes for linear convection–diffusion problems. Numer. Math. 87(3), 457–484 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, X., Zhang, J.: Estimations of the discrete Green’s function of the SDFEM on Shishkin triangular meshes for singularly perturbed problems with characteristic layers. arXiv:1707.02516 (2017)
  15. 15.
    Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707–719 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Roos, H.-G.: Layer-adapted grids for singular perturbation problems. ZAMM Z. Angew. Math. Mech. 78(5), 291–309 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations Vol. 24 of Springer Series in Computational Mathematics, 2nd edn. Springer-Verlag, Berlin (2008)zbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shishkin, G.I.: Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations (In Russian). Second doctoral thesis, Keldysh Institute, Moscow (1990)Google Scholar
  20. 20.
    Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, J., Liu, X.: Analysis of SDFEM on Shishkin triangular meshes and hybrid meshes for problems with characteristic layers. J. Sci. Comput. 68(3), 1299–1316 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, J., Liu, X.: Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers. Adv. Comput. Math. 43(4), 759–775 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, J., Liu, X.: Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with exponential layers BIT Numerical Mathematics (2017)Google Scholar
  25. 25.
    Zhang, J., Liu, X., Yang, M.: Optimal order l 2 error estimate of SDFEM on Shishkin triangular meshes for singularly perturbed convection-diffusion equations. SIAM J. Numer. Anal. 54(4), 2060–2080 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, J., Mei, L.: Pointwise error estimates of the bilinear SDFEM, on Shishkin meshes. Numer. Methods Partial Differential Equations 29(2), 422–440 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems. Math. Comp. 72(243), 1147–1177 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhou, G.: How accurate is the streamline diffusion finite element method. Math. Comp. 66(217), 31–44 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.College of ScienceQilu University of TechnologyJinanChina
  2. 2.School of Mathematics and StatisticsShandong Normal UniversityJinanChina

Personalised recommendations