# 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

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topCheddadi, 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 -

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