A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods

Emmanuel Creusé (1, 2) , Serge Nicaise (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 SIMPAF - SImulations and Modeling for PArticles and Fluids
3 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

We consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posteriori error estimators based on gradient recovery by averaging. It is shown that these estimators give rise to an upper bound where the constant is explicitly known up to some additional terms that guarantees reliability. The lower bound is also established, one being robust when the convection term (or the reaction term) becomes dominant. Moreover, the estimator is asymptotically exact when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimators are confirmed by some numerical tests.

Mots clés: Discontinuous Galerkin finite elements; Convection-diffusion-reaction problems; A posteriori estimator; Discontinuous Galerkin finite elements.

Domaines: Analyse numérique [math.NA]

Emmanuel Creusé, Serge Nicaise. A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods. IMA Journal of Numerical Analysis, 2013, 33 (1), pp.212-241. ⟨10.1093/imanum/drr052⟩. ⟨hal-00531621⟩