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Efficient Equilibrated Flux Reconstruction in High Order Raviart-Thomas Space for Discontinuous Galerkin Methods

  • Igor MozolevskiEmail author
  • Edson Luiz Valmorbida
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

We develop an efficient and computationally cheap method of equilibrated fluxes reconstruction for high-order dG solutions to elliptic problems using a specific computational basis in high order Raviart-Thomas space. The computational basis is designed in such a way that coordinates of equilibrated fluxes with respect to this basis can be easy calculated from the moments of the numerical fluxes of dG method. Some applications of this method in implementation of a posteriori error estimators for elliptic boundary value problems are considered.

Notes

Acknowledgements

The authors gratefully acknowledge the support by CNPq, Brazil, grant 477935/2013-3.

References

  1. 1.
    R.A. Adams, Sobolev Spaces (Academic, New York, 1975)zbMATHGoogle Scholar
  2. 2.
    C. Bahriawati, C. Carstensen, Three matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5(4), 333–361 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44 (Springer, Heidelberg, 2013)Google Scholar
  4. 4.
    D. Braess, V. Pillwein, J. Schöberl, Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198(13–14), 1189–1197 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    D. Braess, T. Fraunholz, R.H.W. Hoppe, An equilibrated a posteriori error estimator for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 52(4), 2121–2136 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    E. Creusé, S. Nicaise, A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods. J. Comput. Appl. Math. 234(10), 2903–2915 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    E. Creusé, S. Nicaise, A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods. IMA J. Numer. Anal. 33(1), 212–241 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques & Applications, vol. 69 (Springer, Berlin, 2011)Google Scholar
  10. 10.
    M. Dubiner, Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    A. Erm, I. Mozolevski, Discontinuous Galerkin method for two-component liquid gas porous media flows. Comput. Geosci. 16, 677–690 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Ern, M. Vohralík, Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. C. R. Math. Acad. Sci. Paris 347(7–8), 441–444 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A. Ern, M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53(2), 1058–1081 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    A. Ern, S. Nicaise, M. Vohralík, An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Math. Acad. Sci. Paris 345(12), 709–712 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    A. Ern, I. Mozolevski, L. Schuh, Accurate velocity reconstruction for discontinuous Galerkin approximations of two-phase porous media flows. C. R. Math. Acad. Sci. Paris 347(9–10), 551–554 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    V. Ervin, Computational bases for RT k and BDM k on triangles. Comput. Math. Appl. 64(8), 2765–2774 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods. Texts in Applied Mathematics, vol. 54 (Springer, New York, 2008). Algorithms, analysis, and applicationsGoogle Scholar
  18. 18.
    P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    K.Y. Kim, A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. Appl. Numer. Math. 57(9), 1065–1080 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    J.M. Melenk, B.I. Wohlmuth, On residual-based a posteriori error estimation in hp-FEM. Adv. Comput. Math. 15(1–4), 311–331 (2002). A posteriori error estimation and adaptive computational methodsGoogle Scholar
  21. 21.
    W.F. Mitchell, A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220, 350–364 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    I. Mozolevski, S. Prudhomme, Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems. Comput. Methods Appl. Mech. Eng. 288, 127–145 (2015)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Federal University of Santa CatarinaFlorianópolisBrazil
  2. 2.Federal University of Technology - ParanáParanáBrazil

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