A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods

Alexandre Ern (1) , Annette Stephansen (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations to advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfürth on continuous finite elements, that the constants are bounded by the square root of the local Péclet number.

Mots clés: Discontinuous Galerkin; weighted interior penalty; a posteriori error estimate; heterogeneous diffusion; advection-diffusion

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Annette Stephansen. A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. 2007. ⟨hal-00171838⟩