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Guaranteed and robust a posteriori error estimatesforsingularly perturbed reaction–diffusion problems

Published online by Cambridge University Press:  30 April 2009

Ibrahim Cheddadi ,
Radek Fučík ,
Mariana I. Prieto  and
Martin Vohralík
Ibrahim Cheddadi
Affiliation:
Univ. Grenoble and CNRS, Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, 38400 Saint Martin d'Hères, France. INRIA Grenoble-Rhône-Alpes, Inovallée, 655 avenue de l'Europe, Montbonnot, 38334 Saint Ismier Cedex, France. ibrahim.cheddadi@imag.fr
Radek Fučík
Affiliation:
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
Mariana I. Prieto
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA, Argentina. mprieto@dm.uba.ar
Martin Vohralík
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. vohralik@ann.jussieu.fr
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Abstract

We derive a posteriori error estimates for singularlyperturbed reaction–diffusion problems which yield a guaranteedupper bound on the discretization error and are fully and easilycomputable. Moreover, they are also locally efficient and robust inthe sense that they represent local lower bounds for the actualerror, up to a generic constant independent in particular of thereaction coefficient. We present our results in the framework ofthe vertex-centered finite volume method but their nature isgeneral for any conforming method, like the piecewise linear finiteelement one. Our estimates are based on a H(div)-conformingreconstruction of the diffusive flux in the lowest-orderRaviart–Thomas–Nédélec space linked with mesh dual to the originalsimplicial one, previously introduced by the last author in thepure diffusion case. They also rely on elaborated Poincaré,Friedrichs, and trace inequalities-based auxiliary estimatesdesigned to cope optimally with the reaction dominance. In order tobring down the ratio of the estimated and actual overall energyerror as close as possible to the optimal value of one,independently of the size of the reaction coefficient, we finallydevelop the ideas of local minimizations of the estimators by localmodifications of the reconstructed diffusive flux. The numericalexperiments presented confirm the guaranteed upper bound,robustness, and excellent efficiency of the derived estimates.

Keywords

Vertex-centered finite volume/finite volume element/box methodsingularly perturbed reaction–diffusion problema posteriori error estimatesguaranteed upper boundrobustness
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 867 - 888
Copyright
© EDP Sciences, SMAI, 2009

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