Stopping Criteria Based on Locally Reconstructed Fluxes

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


We propose stopping criteria for the iterative solution of equations resulting from discretization by conforming, nonconforming, and total discontinuous finite element methods. A simple modification of error estimators based on locally reconstructed fluxes allows to split the estimator into a discretisation-based and an iteration-based part. Comparison of both then leads to stopping criteria which can be used in the framework of an adaptive algorithm.


Posteriori Error Adaptive Mesh Multigrid Method Posteriori Error Estimate Iterative Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Becker, C. Johnson, R. Rannacher, Adaptive error control for multigrid finite element methods. Computing 55, 271–288 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    R. Becker, D. Capatina, R. Luce, Robust local flux reconstruction for various finite element methods, in Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH’ 2013, Lausanne, ed. by A. Abdulle et al. (2014), pp. 65–73. doi:10.1007/978-3-319-10705-9_6Google Scholar
  3. 3.
    D. Braess, J. Schöberl, Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    A. Ern, M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDE’s. SIAM J. Sci. Comput. 35(4), 1761–1791 (2013)CrossRefMathSciNetGoogle Scholar
  5. 5.
    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
  6. 6.
    W. Hackbusch, Multigrid Methods and Applications. Springer Series in Computational Mathematics, vol. 4 (Springer, Berlin, 1985)Google Scholar
  7. 7.
    P. Jirańek, Z. Strakoš, M. Vohralík, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32(3), 1567–1590 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    K.Y. Kim, A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76(257), 43–66 (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    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 (Springer, Berlin, 1977), pp. 292–315Google Scholar
  11. 11.
    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