Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem

Emmanuel Creusé (1) , Serge Nicaise (2)

1 SIMPAF - SImulations and Modeling for PArticles and Fluids
2 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

The paper presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e. elements with very large aspect ratio. Our analysis covers two- and three-dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator.

Mots clés: DG method; Error estimator; Anisotropic solution; Stretched elements; Stokes problem

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Emmanuel Creusé, Serge Nicaise. Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem. Numerical Methods for Partial Differential Equations, 2006, 22 (2), ⟨10.1002/num.20107⟩. ⟨hal-00768690⟩