Skip to main content
SpringerLink
Log in

Two Efficient and Reliable a posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Linear Elliptic Problems on Cartesian Grids

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

Abstract

In this paper, we derive two a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to linear second-order elliptic problems on Cartesian grids. We first prove that the gradient of the LDG solution is superconvergent with order \(p+1\) towards the gradient of Gauss-Radau projection of the exact solution, when tensor product polynomials of degree at most p are used. Then, we prove that the gradient of the actual error can be split into two parts. The components of the significant part can be given in terms of \((p+1)\)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We further develop a postprocessing gradient recovery scheme for the LDG solution. This recovered gradient superconverges to the gradient of the true solution. The order of convergence is proved to be \(p+1\). We use our gradient recovery result to develop a robust recovery-type a posteriori error estimator for the gradient approximation which is based on an enhanced recovery technique. We prove that the proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the \(L^2\)-norm under mesh refinement. The order of convergence is proved to be \(p + 1\). Moreover, the proposed estimators are proved to be asymptotically exact. Finally, we present a local adaptive mesh refinement procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for \(P^p\) polynomials with \(p\ge 1\). We provide several numerical examples illustrating the effectiveness of our procedures.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data Availability Statement

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

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    MATH  Google Scholar 

  2. Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part II: a posteriori error estimation. J. Sci. Comput. 38, 15–49 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Adjerid, S., Baccouch, M.: A superconvergent local discontinuous Galerkin method for elliptic problems. J. Sci. Comput. 52, 113–152 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Adjerid, S., Baccouch, M.: Adaptivity and error estimation for discontinuous Galerkin methods. In: Feng, X., Karakashian, O., Xing, Y. (eds.) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, vol. 157 The IMA Volumes in Mathematics and its Applications, pp. 63–96. Springer, Switzerland (2014)

    MATH  Google Scholar 

  5. Adjerid, S., Chaabane, N.: An improved superconvergence error estimate for the LDG method. Appl. Numer. Math. 98, 122–136 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Adjerid, S., Massey, T.C.: A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 5877–5897 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Ainsworth, M.: A synthesis of a posteriori error estimation techniques for conforming, nonconforming and discontinuous Galerkin finite element methods. In: Recent Advances in Adaptive Computation, Contemp. Math. 383, AMS, pp. 1–14 (2005)

  8. Ainsworth, M.: A Posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45(4), 1777–1798 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Ainsworth, M.: A framework for obtaining guaranteed error bounds for finite element approximations. J. Comput. Appl. Math. 234(9), 2618–2632 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Ainsworth, M., Allendes, A., Barrenechea, G.R., Rankin, R.: On the adaptive selection of the parameter in stabilized finite element approximations. SIAM J. Numer. Anal. 51(3), 1585–1609 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    MATH  Google Scholar 

  12. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Baccouch, M.: A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids. Comput. Math. Appl. 68, 1250–1278 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Baccouch, M.: Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids. Appl. Math. Comput. 265, 144–162 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Baccouch, M.: A Posteriori error analysis of the discontinuous Galerkin method for two-dimensional linear hyperbolic conservation laws on Cartesian grids. J. Sci. Comput. 68(3), 945–974 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Baccouch, M.: A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids. Numer. Methods Partial Differ. Equ. 33(4), 1224–1265 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Baccouch, M.: A recovery-based error estimator for the discontinuous Galerkin method for transient linear hyperbolic conservation laws on Cartesian grids. Int. J. Comput. Methods 14(06), 1750062 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Baccouch, M.: Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation on Cartesian grids. Appl. Numer. Math. 113, 124–155 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Baccouch, M.: Optimal error estimates of the local discontinuous Galerkin method for the two-dimensional sine-Gordon equation on Cartesian grids. Int. J. Numer. Anal. Model. 16, 436–462 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Baccouch, M., Adjerid, S.: Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes. Comput. Methods Appl. Mech. Eng. 200, 162–177 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Baccouch, M., Adjerid, S.: A posteriori local discontinuous Galerkin error estimation for two-dimensional convection-diffusion problems. J. Sci. Comput. 62, 399–430 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser Verlag, Berlin (2003)

    MATH  Google Scholar 

  23. Brezzi, F.: DG methods for elliptic problems. In: Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.) Proceedings of International Symposium on Discontinuous Galerkin Methods Theory, Computation and Applications. Springer, Berlin (2000)

    Google Scholar 

  24. Bustinza, R., Gatica, G., Cockburn, B.: An A Posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22, 147–185 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Castillo, P.: An A Posteriori error estimate for the local discontinuous Galerkin method. J. Sci. Comput. 22, 187–204 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Castillo, P.: A review of the local discontinuous Galerkin (LDG) method applied to elliptic problems. Appl. Numer. Math. 56, 1307–1313 (2006)

    MathSciNet  MATH  Google Scholar 

  27. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)

    MathSciNet  MATH  Google Scholar 

  28. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71, 455–478 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Pub. Co., Amsterdam (1978)

    MATH  Google Scholar 

  30. Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for linearized incompressible fluid flow: a review. Comput. Fluids 34(4–5), 491–506 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)

  34. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    MathSciNet  MATH  Google Scholar 

  35. Dong, B., Shu, C.-W.: Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems. SIAM J. Numer. Anal. 47, 3240–3268 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  37. Gudi, T., Nataraj, N., Pani, A.: An \(hp\)-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type. Math. Comput. 77, 731–756 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Karakashian, O.A., 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)

    MathSciNet  MATH  Google Scholar 

  39. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation, Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos (1991)

  40. Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008)

    MATH  Google Scholar 

  41. Rivière, B., Wheeler, M., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I Comput. Geosci. 3, 337–360 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  43. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Advances in Numerical Mathematics. Wiley, Hoboken (1996)

    MATH  Google Scholar 

  44. Xu, Y., Shu, C.-W.: Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50, 79–104 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Funding

This research was supported by the NASA Nebraska Space Grant (Federal Grant/Award Number 80NSSC20M0112).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE, 68182, USA

    Mahboub Baccouch

Authors
  1. Mahboub Baccouch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahboub Baccouch.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Additional information

Dedicated to Professor Slimane Adjerid on the occasion of his 65th birthday.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baccouch, M. Two Efficient and Reliable a posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Linear Elliptic Problems on Cartesian Grids. J Sci Comput 87, 76 (2021). https://doi.org/10.1007/s10915-021-01497-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01497-w

Keywords

Mathematics Subject Classification

Search

Navigation