Advertisement

Robust Local Flux Reconstruction for Various Finite Element Methods

  • Roland Becker
  • Daniela Capatina
  • Robert LuceEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

We present a uniform approach to local reconstructions of the gradient of primal approximations by conforming, nonconforming and totally discontinuous finite elements of arbitrary order. We start from a hybrid formulation which covers all considered methods and whose Lagrange multipliers approximate the normal fluxes. It turns out that the multipliers can be computed locally and are next used to define local corrections of the flux. We also show that the DG solution and reconstructed flux with stabilisation parameter γ converge uniformly in h with the convergence rate 1∕γ towards the CG or NC ones, depending on the stabilisation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Becker, D. Capatina, J. Joie, Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations. Numer. Methods Part. Differ. Equ. 28(3), 1013–1041 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Braess, J. Schöberl, Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Ern, S. Nicaise, M. Vohralík, An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. A. S. 345(12), 709–712 (2007)zbMATHGoogle Scholar
  4. 4.
    K.Y. Kim, A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76(257), 43–66 (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    R. Luce, B. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42(4), 1394–1414 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22(3), 493–496 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    P.-A. Raviart, J.-M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods: Proceedings of the Conference, Consiglio Naz. delle Ricerche, Rome, 1975 (Springer, Berlin, 1977), pp. 292–315Google Scholar
  8. 8.
    M. Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods. Numer. Math. 111(1), 121–158 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LMAP CNRS UMR 5142Université de Pau, IPRAPauFrance

Personalised recommendations