Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type

Autores
Garau, Eduardo Mario; Morin, Pedro; Zuppa, Carlos
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz
Fil: Garau, Eduardo Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Zuppa, Carlos. Universidad Nacional de San Luis; Argentina
Materia
Adaptive Finite Element Methods
Optimality
Quasilinear Elliptic Equations
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/60505
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone typeGarau, Eduardo MarioMorin, PedroZuppa, CarlosAdaptive Finite Element MethodsOptimalityQuasilinear Elliptic Equationshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and LipschitzFil: Garau, Eduardo Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Zuppa, Carlos. Universidad Nacional de San Luis; ArgentinaGlobal Science Press2012-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/60505Garau, Eduardo Mario; Morin, Pedro; Zuppa, Carlos; Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type; Global Science Press; Numerical Mathematics-theory Methods And Applications; 5; 2; 5-2012; 131-1561004-8979CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.4208/nmtma.2012.m1023info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:10:48Zoai:ri.conicet.gov.ar:11336/60505instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:10:49.278CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
title Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
spellingShingle Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
Garau, Eduardo Mario
Adaptive Finite Element Methods
Optimality
Quasilinear Elliptic Equations
title_short Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
title_full Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
title_fullStr Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
title_full_unstemmed Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
title_sort Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type
dc.creator.none.fl_str_mv Garau, Eduardo Mario
Morin, Pedro
Zuppa, Carlos
author Garau, Eduardo Mario
author_facet Garau, Eduardo Mario
Morin, Pedro
Zuppa, Carlos
author_role author
author2 Morin, Pedro
Zuppa, Carlos
author2_role author
author
dc.subject.none.fl_str_mv Adaptive Finite Element Methods
Optimality
Quasilinear Elliptic Equations
topic Adaptive Finite Element Methods
Optimality
Quasilinear Elliptic Equations
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz
Fil: Garau, Eduardo Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Zuppa, Carlos. Universidad Nacional de San Luis; Argentina
description We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz
publishDate 2012
dc.date.none.fl_str_mv 2012-05
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/11336/60505
Garau, Eduardo Mario; Morin, Pedro; Zuppa, Carlos; Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type; Global Science Press; Numerical Mathematics-theory Methods And Applications; 5; 2; 5-2012; 131-156
1004-8979
CONICET Digital
CONICET
url http://hdl.handle.net/11336/60505
identifier_str_mv Garau, Eduardo Mario; Morin, Pedro; Zuppa, Carlos; Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type; Global Science Press; Numerical Mathematics-theory Methods And Applications; 5; 2; 5-2012; 131-156
1004-8979
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.4208/nmtma.2012.m1023
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Global Science Press
publisher.none.fl_str_mv Global Science Press
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1842270133523841024
score 13.13397

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