Analysis of finite element approximations of stokes equations with nonsmooth data

Autores
Duran, Ricardo Guillermo; Gastaldi, Lucia; Lombardi, Ariel Luis
Año de publicación
2020
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem, improving the error estimates obtained in Cai and Wang [Math. Comp., 78 (2009), pp. 771-787]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.
Fil: Duran, Ricardo Guillermo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Gastaldi, Lucia. Università degli Studi di Brescia; Italia
Fil: Lombardi, Ariel Luis. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina
Materia
A POSTERIORI ERROR ANALYSIS
FINITE ELEMENTS
NONSMOOTH DATA
STOKES EQUATIONS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/151105

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network_name_str CONICET Digital (CONICET)
spelling Analysis of finite element approximations of stokes equations with nonsmooth dataDuran, Ricardo GuillermoGastaldi, LuciaLombardi, Ariel LuisA POSTERIORI ERROR ANALYSISFINITE ELEMENTSNONSMOOTH DATASTOKES EQUATIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem, improving the error estimates obtained in Cai and Wang [Math. Comp., 78 (2009), pp. 771-787]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.Fil: Duran, Ricardo Guillermo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Gastaldi, Lucia. Università degli Studi di Brescia; ItaliaFil: Lombardi, Ariel Luis. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaSociety for Industrial and Applied Mathematics2020-11-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/151105Duran, Ricardo Guillermo; Gastaldi, Lucia; Lombardi, Ariel Luis; Analysis of finite element approximations of stokes equations with nonsmooth data; Society for Industrial and Applied Mathematics; Siam Journal on Numerical Analysis; 58; 6; 12-11-2020; 3309-33310036-14291095-7170CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://epubs.siam.org/doi/10.1137/19M1305872info:eu-repo/semantics/altIdentifier/doi/10.1137/19M1305872info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1912.04962info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:39:18Zoai:ri.conicet.gov.ar:11336/151105instacron: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:34982025-09-29 09:39:19.099CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Analysis of finite element approximations of stokes equations with nonsmooth data
title Analysis of finite element approximations of stokes equations with nonsmooth data
spellingShingle Analysis of finite element approximations of stokes equations with nonsmooth data
Duran, Ricardo Guillermo
A POSTERIORI ERROR ANALYSIS
FINITE ELEMENTS
NONSMOOTH DATA
STOKES EQUATIONS
title_short Analysis of finite element approximations of stokes equations with nonsmooth data
title_full Analysis of finite element approximations of stokes equations with nonsmooth data
title_fullStr Analysis of finite element approximations of stokes equations with nonsmooth data
title_full_unstemmed Analysis of finite element approximations of stokes equations with nonsmooth data
title_sort Analysis of finite element approximations of stokes equations with nonsmooth data
dc.creator.none.fl_str_mv Duran, Ricardo Guillermo
Gastaldi, Lucia
Lombardi, Ariel Luis
author Duran, Ricardo Guillermo
author_facet Duran, Ricardo Guillermo
Gastaldi, Lucia
Lombardi, Ariel Luis
author_role author
author2 Gastaldi, Lucia
Lombardi, Ariel Luis
author2_role author
author
dc.subject.none.fl_str_mv A POSTERIORI ERROR ANALYSIS
FINITE ELEMENTS
NONSMOOTH DATA
STOKES EQUATIONS
topic A POSTERIORI ERROR ANALYSIS
FINITE ELEMENTS
NONSMOOTH DATA
STOKES 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 In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem, improving the error estimates obtained in Cai and Wang [Math. Comp., 78 (2009), pp. 771-787]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.
Fil: Duran, Ricardo Guillermo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Gastaldi, Lucia. Università degli Studi di Brescia; Italia
Fil: Lombardi, Ariel Luis. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina
description In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem, improving the error estimates obtained in Cai and Wang [Math. Comp., 78 (2009), pp. 771-787]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.
publishDate 2020
dc.date.none.fl_str_mv 2020-11-12
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/11336/151105
Duran, Ricardo Guillermo; Gastaldi, Lucia; Lombardi, Ariel Luis; Analysis of finite element approximations of stokes equations with nonsmooth data; Society for Industrial and Applied Mathematics; Siam Journal on Numerical Analysis; 58; 6; 12-11-2020; 3309-3331
0036-1429
1095-7170
CONICET Digital
CONICET
url http://hdl.handle.net/11336/151105
identifier_str_mv Duran, Ricardo Guillermo; Gastaldi, Lucia; Lombardi, Ariel Luis; Analysis of finite element approximations of stokes equations with nonsmooth data; Society for Industrial and Applied Mathematics; Siam Journal on Numerical Analysis; 58; 6; 12-11-2020; 3309-3331
0036-1429
1095-7170
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://epubs.siam.org/doi/10.1137/19M1305872
info:eu-repo/semantics/altIdentifier/doi/10.1137/19M1305872
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1912.04962
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
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
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