Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems

¹ Univ. Grenoble and CNRS, Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, 38400 Saint Martin d'Hères, France.
² INRIA Grenoble-Rhône-Alpes, Inovallée, 655 avenue de l'Europe, Montbonnot, 38334 Saint Ismier Cedex, France. ibrahim.cheddadi@imag.fr
³ Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic. fucik@fjfi.cvut.cz
⁴ Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA, Argentina. mprieto@dm.uba.ar
⁵ UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
⁶ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. vohralik@ann.jussieu.fr

Received: 31 March 2008
Revised: 13 November 2008

Abstract

We derive a posteriori error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart–Thomas–Nédélec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.