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

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. M. Afif Faculté des Sciences-Semlalia, Laboratoire LIBMA, Université Cadi Ayyad, B.P. 2390, Marrakech, Maroc e-mail: [email protected] B. Amaziane Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau et des Pays de l’Adour, av. de l’Université, 64000 Pau, France e-mail: [email protected] G. Kunert IAV GmbH, 09120 Chemnitz, Germany e-mail: [email protected] Z. Mghazli Faculté des Sciences, Laboratoire LIRNE-Equipe EIMA, Université Ibn Tofaïl, B.P. 133, Kénitra, Maroc e-mail: [email protected] S. Nicaise ( ) LAMAV, FR CNRS 2956, Université Lille Nord de France, UVHC, 59313 Valenciennes Cedex 9, France e-mail: [email protected] 184 J Sci Comput (2010) 43: 183–200 Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.