Mixed methods for degenerate elliptic problems and application to fractional Laplacian

Autores
Cejas, María Eugenia; Duran, Ricardo Guillermo; Prieto, Mariana Ines
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A2. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.
Fil: Cejas, María Eugenia. Universidad Nacional de La Plata. Departamento de Matemática, Facultad de Ciencias Exactas. Centro de Matemática La Plata ; Argentina
Fil: Duran, Ricardo Guillermo. 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: Prieto, Mariana Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
Materia
DEGENERATE ELLIPTIC PROBLEMS
FRACTIONAL LAPLACIAN
MIXED FINITE ELEMENTS
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:11336/135120
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Mixed methods for degenerate elliptic problems and application to fractional LaplacianCejas, María EugeniaDuran, Ricardo GuillermoPrieto, Mariana InesDEGENERATE ELLIPTIC PROBLEMSFRACTIONAL LAPLACIANMIXED FINITE ELEMENTShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A2. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.Fil: Cejas, María Eugenia. Universidad Nacional de La Plata. Departamento de Matemática, Facultad de Ciencias Exactas. Centro de Matemática La Plata ; ArgentinaFil: Duran, Ricardo Guillermo. 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: Prieto, Mariana Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaEDP Sciences2021-02-26info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/135120Cejas, María Eugenia; Duran, Ricardo Guillermo; Prieto, Mariana Ines; Mixed methods for degenerate elliptic problems and application to fractional Laplacian; EDP Sciences; ESAIM. Modélisation mathématique et analyse numérique; 55; 26-2-2021; 993-10190764-583X1290-3841CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.esaim-m2an.org/articles/m2an/abs/2021/01/m2an200082/m2an200082.htmlinfo:eu-repo/semantics/altIdentifier/doi/10.1051/m2an/2020068info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1903.05138info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-17T11:11:54Zoai:ri.conicet.gov.ar:11336/135120instacron: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:34982025-09-17 11:11:54.694CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Mixed methods for degenerate elliptic problems and application to fractional Laplacian
title Mixed methods for degenerate elliptic problems and application to fractional Laplacian
spellingShingle Mixed methods for degenerate elliptic problems and application to fractional Laplacian
Cejas, María Eugenia
DEGENERATE ELLIPTIC PROBLEMS
FRACTIONAL LAPLACIAN
MIXED FINITE ELEMENTS
title_short Mixed methods for degenerate elliptic problems and application to fractional Laplacian
title_full Mixed methods for degenerate elliptic problems and application to fractional Laplacian
title_fullStr Mixed methods for degenerate elliptic problems and application to fractional Laplacian
title_full_unstemmed Mixed methods for degenerate elliptic problems and application to fractional Laplacian
title_sort Mixed methods for degenerate elliptic problems and application to fractional Laplacian
dc.creator.none.fl_str_mv Cejas, María Eugenia
Duran, Ricardo Guillermo
Prieto, Mariana Ines
author Cejas, María Eugenia
author_facet Cejas, María Eugenia
Duran, Ricardo Guillermo
Prieto, Mariana Ines
author_role author
author2 Duran, Ricardo Guillermo
Prieto, Mariana Ines
author2_role author
author
dc.subject.none.fl_str_mv DEGENERATE ELLIPTIC PROBLEMS
FRACTIONAL LAPLACIAN
MIXED FINITE ELEMENTS
topic DEGENERATE ELLIPTIC PROBLEMS
FRACTIONAL LAPLACIAN
MIXED FINITE ELEMENTS
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 analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A2. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.
Fil: Cejas, María Eugenia. Universidad Nacional de La Plata. Departamento de Matemática, Facultad de Ciencias Exactas. Centro de Matemática La Plata ; Argentina
Fil: Duran, Ricardo Guillermo. 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: Prieto, Mariana Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
description We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A2. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.
publishDate 2021
dc.date.none.fl_str_mv 2021-02-26
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/11336/135120
Cejas, María Eugenia; Duran, Ricardo Guillermo; Prieto, Mariana Ines; Mixed methods for degenerate elliptic problems and application to fractional Laplacian; EDP Sciences; ESAIM. Modélisation mathématique et analyse numérique; 55; 26-2-2021; 993-1019
0764-583X
1290-3841
CONICET Digital
CONICET
url http://hdl.handle.net/11336/135120
identifier_str_mv Cejas, María Eugenia; Duran, Ricardo Guillermo; Prieto, Mariana Ines; Mixed methods for degenerate elliptic problems and application to fractional Laplacian; EDP Sciences; ESAIM. Modélisation mathématique et analyse numérique; 55; 26-2-2021; 993-1019
0764-583X
1290-3841
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://www.esaim-m2an.org/articles/m2an/abs/2021/01/m2an200082/m2an200082.html
info:eu-repo/semantics/altIdentifier/doi/10.1051/m2an/2020068
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1903.05138
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
dc.publisher.none.fl_str_mv EDP Sciences
publisher.none.fl_str_mv EDP Sciences
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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score 13.001348

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