Métodos numéricos para problemas no locales de evolución

Autores
Mastroberti Bersetche, Francisco Vicente
Año de publicación
2019
Idioma
inglés
Tipo de recurso
tesis doctoral
Estado
versión publicada
Colaborador/a o director/a de tesis
Acosta Rodriguez, Gabriel
Descripción
This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.
The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documented
Fil: Mastroberti Bersetche, Francisco Vicente. 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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Materia
Laplaciano Fraccionario
Derivada de Caputo
Método de Elementos Finitos
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:11336/80087

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spelling Métodos numéricos para problemas no locales de evoluciónNumerical methods for non-local evolution problemsMastroberti Bersetche, Francisco VicenteLaplaciano FraccionarioDerivada de CaputoMétodo de Elementos Finitoshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documentedFil: Mastroberti Bersetche, Francisco Vicente. 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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaAcosta Rodriguez, Gabriel2019-03-06info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_db06info:ar-repo/semantics/tesisDoctoralapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/80087Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://cms.dm.uba.ar/academico/carreras/doctorado/Tesis%20mastroberti.pdfinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:43:58Zoai:ri.conicet.gov.ar:11336/80087instacron: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 09:43:58.789CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Métodos numéricos para problemas no locales de evolución
Numerical methods for non-local evolution problems
title Métodos numéricos para problemas no locales de evolución
spellingShingle Métodos numéricos para problemas no locales de evolución
Mastroberti Bersetche, Francisco Vicente
Laplaciano Fraccionario
Derivada de Caputo
Método de Elementos Finitos
title_short Métodos numéricos para problemas no locales de evolución
title_full Métodos numéricos para problemas no locales de evolución
title_fullStr Métodos numéricos para problemas no locales de evolución
title_full_unstemmed Métodos numéricos para problemas no locales de evolución
title_sort Métodos numéricos para problemas no locales de evolución
dc.creator.none.fl_str_mv Mastroberti Bersetche, Francisco Vicente
author Mastroberti Bersetche, Francisco Vicente
author_facet Mastroberti Bersetche, Francisco Vicente
author_role author
dc.contributor.none.fl_str_mv Acosta Rodriguez, Gabriel
dc.subject.none.fl_str_mv Laplaciano Fraccionario
Derivada de Caputo
Método de Elementos Finitos
topic Laplaciano Fraccionario
Derivada de Caputo
Método de Elementos Finitos
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.
The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documented
Fil: Mastroberti Bersetche, Francisco Vicente. 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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
description This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.
publishDate 2019
dc.date.none.fl_str_mv 2019-03-06
dc.type.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_db06
info:ar-repo/semantics/tesisDoctoral
format doctoralThesis
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/80087
Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019
CONICET Digital
CONICET
url http://hdl.handle.net/11336/80087
identifier_str_mv Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://cms.dm.uba.ar/academico/carreras/doctorado/Tesis%20mastroberti.pdf
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.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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