A Posteriori Error Estimation for a Finite Volume Discretization on Anisotropic Meshes

Authors:
M. Afif;B. Amaziane;G. Kunert;Z. Mghazli;S. Nicaise
Affiliations:
Faculté des Sciences-Semlalia, Laboratoire LIBMA, Université Cadi Ayyad, Marrakech, Maroc;Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau et des Pays de l'Adour, Pau, France 64000;IAV GmbH, Chemnitz, Germany 09120;Faculté des Sciences, Laboratoire LIRNE-Equipe EIMA, Université Ibn Tofaïl, Kénitra, Maroc;LAMAV, FR CNRS 2956, Université Lille Nord de France, UVHC, Valenciennes Cedex 9, France 59313
Venue:
Journal of Scientific Computing
Year:
2010

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Abstract

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.