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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.
Mots-clés : vertex-centered finite volume/finite volume element/box method, singularly perturbed reaction-diffusion problem, a posteriori error estimates, guaranteed upper bound, robustness
@article{M2AN_2009__43_5_867_0,
author = {Cheddadi, Ibrahim and Fu\v{c}{\'\i}k, Radek and Prieto, Mariana I. and Vohral{\'\i}k, Martin},
title = {Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {867--888},
publisher = {EDP-Sciences},
volume = {43},
number = {5},
year = {2009},
doi = {10.1051/m2an/2009012},
mrnumber = {2559737},
language = {en},
url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009012/}
}
TY - JOUR AU - Cheddadi, Ibrahim AU - Fučík, Radek AU - Prieto, Mariana I. AU - Vohralík, Martin TI - Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2009 SP - 867 EP - 888 VL - 43 IS - 5 PB - EDP-Sciences UR - https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009012/ DO - 10.1051/m2an/2009012 LA - en ID - M2AN_2009__43_5_867_0 ER -
%0 Journal Article %A Cheddadi, Ibrahim %A Fučík, Radek %A Prieto, Mariana I. %A Vohralík, Martin %T Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2009 %P 867-888 %V 43 %N 5 %I EDP-Sciences %U https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009012/ %R 10.1051/m2an/2009012 %G en %F M2AN_2009__43_5_867_0
Cheddadi, Ibrahim; Fučík, Radek; Prieto, Mariana I.; Vohralík, Martin. Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 5, pp. 867-888. doi : 10.1051/m2an/2009012. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009012/
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