Skip to main content
SpringerLink
Log in

A unified a posteriori error estimate of local discontinuous Galerkin approximations for reactive transport problems

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

To solve reactive transport problems in porous media, local discontinuous Galerkin (LDG) approximations are investigated. Based on the duality technique and the residual error notations, a unified a posteriori error estimate in L 2(L 2) norm is obtained, which is usually used for guiding anisotropic and dynamic mesh adaptivity.

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. V. Aizinger, C. Dawson, B. Cockburn and P. Castillo, The local discontinuous Galerkin method for contaminant transport, Adv. Water Resources, 24(1) (2000), 73–87.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Babuška and M. Suri, The hp version of the finite element method with quasi-uniform meshes, RAIRO Model. Math. Anal. Numer., 21 (1987), 199–238.

    MATH  MathSciNet  Google Scholar 

  4. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier-stokes equations, J. Comp. Phys., 131 (1997), 267–279.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Bustinza, G. N. Gatica and B. Cockburn, An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems, J. Sci. Comp., 22–23(1) (2005), 147–185.

    Article  MathSciNet  Google Scholar 

  6. P. Castillo, B. Cockburn, I. Perugia and D. Schotzau, Local eiscontinuous Galerkin method for elliptic problems, Comm. Numer. Meth. Engr., 18(1) (2002), 69–75.

    Article  MATH  Google Scholar 

  7. P. Castillo, An a posteriori error estimate for the local discontinuous Galerkin method, J. Sci. Comp., 22–23(1) (2005), 187–204.

    Article  Google Scholar 

  8. P. Castillo and E. Velazquez, An adaptive strategy for the local discontinuous Galerkin method applied to porous media problems, Computer Aided Civil and Infrastructure Engineering, 28 (2008), 238–252.

    Google Scholar 

  9. P. Castillo, Performance of discontinuous Galerkin methods for elliptic PDEs, SIAM J. Sci. Comp., 24(2) (2002), 524–547.

    Article  MATH  Google Scholar 

  10. 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., 43(10) (2007), 757–770.

    Article  MathSciNet  Google Scholar 

  11. B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, In Proceedings of the 10th Conference on the Mathematics of Finite Elements and Applications, edited by J. Whiteman, Elsevier, (2000), 225–238.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservations laws II: Genaral framework, Math. Comp., 52 (1989), 411–435.

    MATH  MathSciNet  Google Scholar 

  14. B. Cockburn, S. Hou and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservations laws IV: The multidimensional case, Math. Comp., 54 (1990), 545–581.

    MATH  MathSciNet  Google Scholar 

  15. C. Dawson and J. Proft, A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations, Numer. Methods Part. Diff. Eqns., 17(6) (2001), 545–564.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. O. Ladyzhenskaya, The boundary value problems of mathematical physics, Springer Verlag, New York, 1985.

    Book  MATH  Google Scholar 

  18. 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., 46 (2003), 141–163.

    Article  MATH  Google Scholar 

  19. B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and implementation, SIAM: philadelphia, PA, USA, 2008.

    Book  Google Scholar 

  20. 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., 22 (2005), 511–540.

    MathSciNet  Google Scholar 

  21. S. Sun and M. F. Wheeler, A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems, Comp. Meth. Appl. Mech. Engrg., 195(7–8) (2006), 632–652.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Jiming Yang

  2. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, 411105, Hunan, China

    Jiming Yang

Authors
  1. Jiming Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiming Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J. A unified a posteriori error estimate of local discontinuous Galerkin approximations for reactive transport problems. Indian J Pure Appl Math 46, 759–772 (2015). https://doi.org/10.1007/s13226-015-0160-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-015-0160-5

Key words

Search

Navigation