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

Ibrahim Cheddadi; Radek Fučík; Mariana I. Prieto; Martin Vohralík

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 5, page 867-888
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Cheddadi, Ibrahim, et al. "Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 867-888. <http://eudml.org/doc/250601>.

@article{Cheddadi2009,
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. },
author = {Cheddadi, Ibrahim, Fučík, Radek, Prieto, Mariana I., Vohralík, Martin},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Vertex-centered finite volume/finite volume element/box method; singularly perturbed reaction–diffusion problem; a posteriori error estimates; guaranteed upper bound; robustness; vertex-centered finite volume/finite volume element/box method; singular perturbation; reaction-diffusion problem; a posteriori error estimates; lowest-order Raviart-Thomas-Nédélec space; numerical experiments; efficiency},
language = {eng},
month = {4},
number = {5},
pages = {867-888},
publisher = {EDP Sciences},
title = {Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems},
url = {http://eudml.org/doc/250601},
volume = {43},
year = {2009},
}

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
DA - 2009/4//
PB - EDP Sciences
VL - 43
IS - 5
SP - 867
EP - 888
AB - 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.
LA - eng
KW - Vertex-centered finite volume/finite volume element/box method; singularly perturbed reaction–diffusion problem; a posteriori error estimates; guaranteed upper bound; robustness; vertex-centered finite volume/finite volume element/box method; singular perturbation; reaction-diffusion problem; a posteriori error estimates; lowest-order Raviart-Thomas-Nédélec space; numerical experiments; efficiency
UR - http://eudml.org/doc/250601
ER -

References

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