An A Posteriori Error Estimator for a Finite Volume Discretization of the Two-phase Flow

  • Daniele A. Di PietroEmail author
  • Martin Vohralík
  • Carole Widmer
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 4)


We derive a posteriori error estimates for a multi-point finite volume discretization of the two-phase Darcy problem. The proposed estimators yield a fully computable upper bound for the selected error measure. The estimate also allows to distinguish, estimate separately, and compare the linearization and algebraic errors and the time and space discretization errors. This enables, in particular, to design a discretization algorithm so that all the sources of error are properly balanced. Namely, the linear and nonlinear solvers can be stopped as soon as the algebraic and linearization errors drop to the level at which they do not affect to the overall error. This can lead to significant computational savings, since performing an excessive number of unnecessary iterations can be avoided. Similarly, the errors in space and in time can be equilibrated by time step and local mesh adaptivity.


Finite volumes a posteriori error estimates darcy model fully computable upperbound twophase flow 


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  1. 1.
    L. AGéLAS, D.A. DI PIETRO, AND R. MASSON, A symmetric and coercive finite volume scheme for multiphase porous media flow with applications in the oil industry, (2008), pp. 35–52.Google Scholar
  2. 2.
    C. CANCèS AND M. VOHRALíK, A posteriori error estimtate for immiscible incompressible two-phase flows. In preparation, 2011.Google Scholar
  3. 3.
    Z. CHEN, G. HUAN, AND Y. MA, Computational methods for multiphase flows in porous media, Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.Google Scholar
  4. 4.
    L. EL ALAOUI, A. ERN, AND M. VOHRALíK, Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems, Comput. Methods Appl. Mech. Engrg., (2010). DOI 10.1016/j.cma.2010.03.024.Google Scholar
  5. 5.
    A. ERN AND M. VOHRALíK, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal., 48 (2010), pp. 198–223.Google Scholar
  6. 6.
    P. JIRáNEK, Z. STRAKOš, AND M. VOHRALíK, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput., 32 (2010), pp. 1567–1590.Google Scholar
  7. 7.
    C. MARLES, Cours de production, Tome 4, Technip, 1984.Google Scholar
  8. 8.
    R. VERFüRTH, Robust a posteriori error estimates for nonstationary convection-diffusion equations, SIAM J. Numer. Anal., 43 (2005), pp. 1783–1802.Google Scholar
  9. 9.
    M. VOHRALíK, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. In preparation, 2011.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniele A. Di Pietro
    • 1
    Email author
  • Martin Vohralík
    • 2
    • 3
  • Carole Widmer
    • 1
  1. 1.IFP Energies nouvellesRueil-MalmaisonFrance
  2. 2.UMR 7598, Laboratoire J.-L. LionsUPMC Univ. Paris 06ParisFrance
  3. 3.UMR 7598, Laboratoire J.-L. LionsCNRSParisFrance

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