Skip to main content
SpringerLink
Log in

A posteriori error analysis for a fully discrete discontinuous Galerkin approximation to a kind of reactive transport problems

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

In order to obtain an expected numerical solution, a fully discrete discontinuous Galerkin method is applied to a kind of reactive transport problems in two dimension. That is to say, the space variable is discretized with the symmetric interior penalty Galerkin method (SIPG), and the time variable is done with the backward Euler method. Making use of the duality technique, hp approximation properties and the interpolation theory, a residual-type a posteriori error estimation is achieved, which can be used for adaptivity. Compared with the analyses of semi-discretization, the current presentation is more challenging and more significant.

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. J. Yang and Y. Chen, A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations, J. Comput. Math., 2006, 24(3): 425–434.

    MathSciNet  MATH  Google Scholar 

  2. H. Chen and Y. Chen, A combined mixed finite element and discontinuous Galerkin method for compressible miscible displacement problem, Natur. Sci. J. Xiangtan Univ., 2004, 26(2): 119–126.

    MathSciNet  MATH  Google Scholar 

  3. J. T. Oden, I. Babuška, and C. E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comp. Phys., 1998, 146: 491–516.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Sun, Discontinuous Galerkin methods for reactive transport in porous media, Ph. D. thesis, The university of Texas at Austin, 2003.

  6. M. F. Wheeler, S. Sun, O. Eslinger, and B. Rivière, Discontinuous Galerkin method for modeling flow and reactive transport in porous media, W. Wendland, ed., Analysis and Simulation of Multified Problem, Springer-Verlag, 2003.

  7. Y. Chen and J. Yang, A posteriori error estimation for a fully discrete discontinuous Galerkin approximation to a kind of singularly perturbed problems, Finite Elem. Anal. Des., 2007, 43(10): 757–770.

    Article  MathSciNet  Google Scholar 

  8. J. Yang and Y. Chen, A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems, Sci. China Math., 2010, 53(10): 2679–2696.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Yang and Y. Chen, A priori error estimates of a combined mixed finite element and discontinuous Galerkin method for compressible miscible displacement with molecular diffusion and dispersion, J. Comp. Math., 2011, 29(1): 93–110.

    Google Scholar 

  10. C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engry., 1993, 107: 117–129.

    Article  MATH  Google Scholar 

  11. M. Larson and T. Barth, A posteriori error estimation for adaptive discontinuous Galerkin approximations of hyperbolic systems, Discontinuous Galerkin Methods, B. Cockburn, G. Karniadakis, and C. Shu, eds., Springer-Verlag in Berlin, 2000.

  12. B. Rivière and M. F. Wheeler, A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems, Comp. Math. Appl., 2003, 46: 141–163.

    Article  MATH  Google Scholar 

  13. A. Ern and J. Proft, A posteriori discontinuous Galerkin error estimates for transient convection-diffusion equations, Appl. Math. Letters, 2005, 18: 833–841.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Sun and M. F. Wheeler, A posteriori error analyses for symmetric discontinuous Galerkin approximations of reactive transport problems, TICAM report 03-19, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, 2003.

    Google Scholar 

  15. S. Sun and M. F. Wheeler, L 2(H 1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. Sci. Comp., 2005, 22: 511–540.

    Article  MathSciNet  Google Scholar 

  16. C. Sun and S. Qin, The full discrete discontinuous finite element analysis for first-order linear hyperbolic equation, J. Comp. Math., 1999, 17(1): 97–112.

    MathSciNet  MATH  Google Scholar 

  17. I. Babuška and M. Suri, The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 1987, 24: 750–776.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Hunan Institute of Engineering, Xiangtan, 411104, China

    Jiming Yang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yanping Chen

Authors
  1. Jiming Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiming Yang.

Additional information

The first author is supported by Hunan Provincial Natural Science Foundation of China under Grant No. 10JJ3021, Scientific Research Fund of Hunan Provincial Education Department under Grant No. 11B032, the Planned Science and Technology Project of Hunan Province and Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

This paper was recommended for publication by Editor Ningning YAN.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, J., Chen, Y. A posteriori error analysis for a fully discrete discontinuous Galerkin approximation to a kind of reactive transport problems. J Syst Sci Complex 25, 398–409 (2012). https://doi.org/10.1007/s11424-011-9338-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-9338-z

Key words

Search

Navigation