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

Autores
Cheddadi, I.; Fučík, R.; Prieto, M.I.; Vohralík, M.
Año de publicación
2009
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
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. © 2009 EDP Sciences SMAI.
Fil:Prieto, M.I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fuente
Math. Model. Numer. Anal. 2009;43(5):867-888
Materia
A posteriori error estimates
Guaranteed upper bound
Robustness
Singularly perturbed reaction-diffusion problem
Vertex-centered finite volume/finite volume element/box method
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by/2.5/ar
Repositorio
Biblioteca Digital (UBA-FCEN)
Institución
Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
OAI Identificador
paperaa:paper_0764583X_v43_n5_p867_Cheddadi

id BDUBAFCEN_01ff1747f70fe6cc59b3bd0541275be1
oai_identifier_str paperaa:paper_0764583X_v43_n5_p867_Cheddadi
network_acronym_str BDUBAFCEN
repository_id_str 1896
network_name_str Biblioteca Digital (UBA-FCEN)
spelling Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problemsCheddadi, I.Fučík, R.Prieto, M.I.Vohralík, M.A posteriori error estimatesGuaranteed upper boundRobustnessSingularly perturbed reaction-diffusion problemVertex-centered finite volume/finite volume element/box methodWe 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. © 2009 EDP Sciences SMAI.Fil:Prieto, M.I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.2009info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://hdl.handle.net/20.500.12110/paper_0764583X_v43_n5_p867_CheddadiMath. Model. Numer. Anal. 2009;43(5):867-888reponame:Biblioteca Digital (UBA-FCEN)instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesinstacron:UBA-FCENenginfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/ar2025-09-29T13:42:55Zpaperaa:paper_0764583X_v43_n5_p867_CheddadiInstitucionalhttps://digital.bl.fcen.uba.ar/Universidad públicaNo correspondehttps://digital.bl.fcen.uba.ar/cgi-bin/oaiserver.cgiana@bl.fcen.uba.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:18962025-09-29 13:42:56.496Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
title Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
spellingShingle Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
Cheddadi, I.
A posteriori error estimates
Guaranteed upper bound
Robustness
Singularly perturbed reaction-diffusion problem
Vertex-centered finite volume/finite volume element/box method
title_short Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
title_full Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
title_fullStr Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
title_full_unstemmed Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
title_sort Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
dc.creator.none.fl_str_mv Cheddadi, I.
Fučík, R.
Prieto, M.I.
Vohralík, M.
author Cheddadi, I.
author_facet Cheddadi, I.
Fučík, R.
Prieto, M.I.
Vohralík, M.
author_role author
author2 Fučík, R.
Prieto, M.I.
Vohralík, M.
author2_role author
author
author
dc.subject.none.fl_str_mv A posteriori error estimates
Guaranteed upper bound
Robustness
Singularly perturbed reaction-diffusion problem
Vertex-centered finite volume/finite volume element/box method
topic A posteriori error estimates
Guaranteed upper bound
Robustness
Singularly perturbed reaction-diffusion problem
Vertex-centered finite volume/finite volume element/box method
dc.description.none.fl_txt_mv 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. © 2009 EDP Sciences SMAI.
Fil:Prieto, M.I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
description 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. © 2009 EDP Sciences SMAI.
publishDate 2009
dc.date.none.fl_str_mv 2009
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.12110/paper_0764583X_v43_n5_p867_Cheddadi
url http://hdl.handle.net/20.500.12110/paper_0764583X_v43_n5_p867_Cheddadi
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/2.5/ar
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/2.5/ar
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv Math. Model. Numer. Anal. 2009;43(5):867-888
reponame:Biblioteca Digital (UBA-FCEN)
instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron:UBA-FCEN
reponame_str Biblioteca Digital (UBA-FCEN)
collection Biblioteca Digital (UBA-FCEN)
instname_str Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron_str UBA-FCEN
institution UBA-FCEN
repository.name.fl_str_mv Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
repository.mail.fl_str_mv ana@bl.fcen.uba.ar
_version_ 1844618735514550272
score 13.070432