Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems

  • Authors:

  • Alexandre Ern;Annette F. Stephansen;Martin Vohralík

  • Affiliations:

  • Université Paris-Est, CERMICS, Ecole des Ponts, 77455 Marne la Vallée cedex 2, France;Université Paris-Est, CERMICS, Ecole des Ponts, 77455 Marne la Vallée cedex 2, France and ANDRA, Parc de la Croix-Blanche, 1-7 rue Jean Monnet, 92298 Chítenay-Malabry, France and Ce ...;UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

Quantified Score

Hi-index 7.29

Visualization

Abstract

We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div,@W)-conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Peclet and Damkohler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.