Skip to main content
SpringerLink
Log in

Error analysis of discontinuous Galerkin methods on layer adapted meshes for two dimensional turning point problem

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

A class of two-dimensional singularly perturbed convection–diffusion problems with turning points is studied in this work. Since the turning point takes place inside the domain, the problem usually exhibits a weak interior layer of cusp-type. The inclusion of parabolic boundary layers in the solution is caused by the presence of a small perturbation parameter. Two methods are designed to outperform the error estimation: the non-symmetric discontinuous Galerkin technique with an interior penalty (NIPG) and the symmetric discontinuous Galerkin method with an interior penalty (SIPG). To circumvent the layering effect in domain discretization, the traditional Shishkin mesh is used. Uniform error estimates are derived in the respective norms of order \(\mathcal {O}(N^{-1} \ln ^{3/2} N)\) and \(\mathcal {O}(N^{-1} \ln ^{2} N)\) for the NIPG and the SIPG methods, respectively. Numerical experiments are carried out to validate theoretical findings.

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
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability statements

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

References

  1. AbdulRidha, M.W., Kashkool, H.A.: The error analysis for the discontinuous Galerkin finite element method of the convection–diffusion problem. J. Basrah Res. Sci. 45(2), (2019)

  2. Çörekli, Ç.: The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions. (2022)

  3. Dillon, R., Maini, P.K., Othmer, H.G.: Pattern formation in generalized turing systems: I Steady-state patterns in systems with mixed boundary conditions. J. Math. Biol. 32, 345–393 (1994)

    Article  MathSciNet  Google Scholar 

  4. Dobrowolski, M., Roos, H.G.: A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. Zeitschrift für Analysis und Ihre Anwendungen 16(4), 1001–1012 (1997)

    Article  MathSciNet  Google Scholar 

  5. Franz, S., Linß, T.: Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection–diffusion problem with characteristic layers. Numer. Methods Partial Differ. Equ. Int. J. 24(1), 144–164 (2008)

  6. Houston, P., Schwab, C., Süli, E.: Discontinuous HP-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002)

    Article  MathSciNet  Google Scholar 

  7. Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion problems. Tech. Rep. SIAM J. Numer. Anal. 39(6), 2133–2163 (2000)

    Article  MathSciNet  Google Scholar 

  8. Lazarov, R.D., Tobiska, L., Vassilevski, P.S.: Streamline diffusion least-squares mixed finite element methods for convection-diffusion problems. East West J. Numer. Math. 5, 249–264 (1997)

    MathSciNet  Google Scholar 

  9. Li, J.: Uniform convergence of discontinuous finite element methods for singularly perturbed reaction-diffusion problems. Comput. Math. Appl. 44(1–2), 231–240 (2002)

    Article  MathSciNet  Google Scholar 

  10. Li, J., Wheeler, M.F.: Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids. SIAM J. Numer. Anal. 38(3), 770–798 (2000)

    Article  MathSciNet  Google Scholar 

  11. Lin, R.: Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities. Numer. Math. 112(2), 295–318 (2009)

    Article  MathSciNet  Google Scholar 

  12. Melenk, J.M., Schwab, C.H.: Analytic regularity for a singularly perturbed problem. SIAM J. Math. Anal. 30(2), 379–400 (1999)

    Article  MathSciNet  Google Scholar 

  13. Oden, J.T., Babuska, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)

    Article  MathSciNet  Google Scholar 

  14. Prudhomme, S., Pascal, F., Oden, J.T., Romkes, A.: Review of a priori error estimation for discontinuous Galerkin methods. (2000)

  15. Ranjan, K.R., Gowrisankar, S.: Uniformly convergent NIPG method for singularly perturbed convection diffusion problem on Shishkin type meshes. Appl. Numer. Math. 179(4), 125–148 (2022)

    Article  MathSciNet  Google Scholar 

  16. Ranjan, K.R., Gowrisankar, S.: NIPG method on Shishkin mesh for singularly perturbed convection-diffusion problem with discontinuous source term. Int. J. Comput. Methods 20(2), 2250048 (2023)

    Article  MathSciNet  Google Scholar 

  17. Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902–931 (2001)

    Article  MathSciNet  Google Scholar 

  18. Roos, Hans-Görg.: Layer-adapted grids for singular perturbation problems. ZAMM-J. Appl. Math. Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Appl. Math. Mech. 78(5), 291–309 (1998)

  19. Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems. Springer, New York (2008)

    Google Scholar 

  20. Roos, H.G., Zarin, H.: The discontinuous Galerkin finite element method for singularly perturbed problems. In: Challenges in scientific computing-CISC 2002, pp. 246–267. Springer, New York (2003)

  21. Roos, H.G., Zarin, H.: A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes. Numer. Methods Part. Differ. Equ. Int. J. 23(6), 1560–1576 (2007)

  22. Schatz, A.H., Wahlbin, L.B.: On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions. Math. Comput. 40(161), 47–89 (1983)

    Article  MathSciNet  Google Scholar 

  23. Shishkin, G.I.: Discrete approximation of singularly perturbed elliptic and parabolic equations. Russian Acad. Sci. Ural Sect. Ekaterinburg, 269, (1992)

  24. Singh, S., Kumar, D., Aguiar, J.V.: A robust numerical technique for weakly coupled system of parabolic singularly perturbed reaction-diffusion equations. J. Math. Chem. 61(6), 1313–1350 (2023)

    Article  MathSciNet  Google Scholar 

  25. Song, L.: A fully discrete SIPG method for solving two classes of vortex dominated flows. Vortex Dyn. Theor. Appl. 81, (2020)

  26. Sun, G., Stynes, M.: Finite element methods on piecewise equidistant meshes for interior turning point problems. Numer. Algorithms 8(1), 111–129 (1994)

    Article  MathSciNet  Google Scholar 

  27. Wahlbin, L.B.: Local Behaviour in Finite Element Methods 2, 353–522 (1991)

    Google Scholar 

  28. Yang, Y., Zhu, P.: Discontinuous Galerkin methods with interior penalties on graded meshes for 2d singularly perturbed convection-diffusion problems. Appl. Numer. Math. 111, 36–48 (2017)

    Article  MathSciNet  Google Scholar 

  29. Yücel, H., Heinkenschloss, M., Karasözen, B.: Distributed optimal control of diffusion–convection–reaction equations using discontinuous Galerkin methods. In: Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, PP. 389–397. Springer, New York (2012)

  30. Zarin, H.: Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers. J. Comput. Appl. Math. 231(2), 626–636 (2009)

    Article  MathSciNet  Google Scholar 

  31. Zarin, H., Gordic, S.: Numerical solving of singularly perturbed boundary value problems with discontinuities. Novi Sad J. Math. 42(1), 131–145 (2012)

    MathSciNet  Google Scholar 

  32. 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  Google Scholar 

  33. Zhang, J., Ma, X., Lv, Y.: Finite element method on Shishkin mesh for a singularly perturbed problem with an interior layer. Appl. Math. Lett. 121, 107509 (2021)

    Article  MathSciNet  Google Scholar 

  34. Zhang, Z.: Superconvergent approximation of singularly perturbed problems. In: Technical Report 97-ZZ2, Department of Mathematics and Statistics, Texas Tech University, (1997)

  35. Zhou, G.: How accurate is the streamline diffusion method. Preprint 95: 22, (1995)

Download references

Acknowledgements

The authors wish to thank the anonymous referees for their remarks that contributed to improve the presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Patna, Patna, 800005, India

    Kumar Rajeev Ranjan & S. Gowrisankar

Authors
  1. Kumar Rajeev Ranjan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Gowrisankar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both the authors have participated in (a) conception and analysis; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. Both the authors have read and agreed to the final submitted version of the manuscript.

Corresponding author

Correspondence to S. Gowrisankar.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Availability of supporting data

Not applicable.

Additional information

Publisher's Note

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

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

Ranjan, K.R., Gowrisankar, S. Error analysis of discontinuous Galerkin methods on layer adapted meshes for two dimensional turning point problem. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02054-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12190-024-02054-y

Keywords

Search

Navigation