On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite e...

Descripción completa

Detalles Bibliográficos
Autores: García-Archilla, Bosco, John, Volker, Novo Martín, Julia
Tipo de recurso: artículo
Fecha de publicación:2021
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/700580
Acceso en línea:http://hdl.handle.net/10486/700580
https://dx.doi.org/10.1016/j.cma.2021.114032
Access Level:acceso abierto
Palabra clave:Convection-dominated regime
Convection–diffusion equations
Convergence of the error of the kinetic energy
Finite element methods
Incompressible Navier–Stokes equations
Robust error bounds
Matemáticas
id ES_25047bc0d8dcef66f7ef8def3a3ebe8e
oai_identifier_str oai:repositorio.uam.es:10486/700580
network_acronym_str ES
network_name_str España
repository_id_str
spelling On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flowsGarcía-Archilla, BoscoJohn, VolkerNovo Martín, JuliaConvection-dominated regimeConvection–diffusion equationsConvergence of the error of the kinetic energyFinite element methodsIncompressible Navier–Stokes equationsRobust error boundsMatemáticasThe kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite element error analysis for the velocity error in L∞(0,T;L2(Ω)) is reviewed, where T is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order r−1, r, and r+1∕2 for the velocity error in L∞(0,T;L2(Ω)). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still openThe research of Bosco García-Archilla has been supported by Spanish MICIU under grant PGC2018-096265-B-I00 and MICI under grant PID2019-104141GB-I00. The Research of Julia Novo has been supported by Spanish MICI under grants PID2019-104141GB-I00 and VA169P20 (Junta de Castilla y Leon, ES) cofinanced by FEDER fundsElsevierDepartamento de MatemáticasFacultad de Ciencias20212021-07-24research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/700580https://dx.doi.org/10.1016/j.cma.2021.114032reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/7005802026-06-23T12:46:27Z
dc.title.none.fl_str_mv On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
title On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
spellingShingle On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
García-Archilla, Bosco
Convection-dominated regime
Convection–diffusion equations
Convergence of the error of the kinetic energy
Finite element methods
Incompressible Navier–Stokes equations
Robust error bounds
Matemáticas
title_short On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
title_full On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
title_fullStr On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
title_full_unstemmed On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
title_sort On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
dc.creator.none.fl_str_mv García-Archilla, Bosco
John, Volker
Novo Martín, Julia
author García-Archilla, Bosco
author_facet García-Archilla, Bosco
John, Volker
Novo Martín, Julia
author_role author
author2 John, Volker
Novo Martín, Julia
author2_role author
author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
dc.subject.none.fl_str_mv Convection-dominated regime
Convection–diffusion equations
Convergence of the error of the kinetic energy
Finite element methods
Incompressible Navier–Stokes equations
Robust error bounds
Matemáticas
topic Convection-dominated regime
Convection–diffusion equations
Convergence of the error of the kinetic energy
Finite element methods
Incompressible Navier–Stokes equations
Robust error bounds
Matemáticas
description The kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite element error analysis for the velocity error in L∞(0,T;L2(Ω)) is reviewed, where T is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order r−1, r, and r+1∕2 for the velocity error in L∞(0,T;L2(Ω)). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open
publishDate 2021
dc.date.none.fl_str_mv 2021
2021-07-24
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/700580
https://dx.doi.org/10.1016/j.cma.2021.114032
url http://hdl.handle.net/10486/700580
https://dx.doi.org/10.1016/j.cma.2021.114032
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Biblos-e Archivo. Repositorio Institucional de la UAM
instname:Universidad Autónoma de Madrid
instname_str Universidad Autónoma de Madrid
reponame_str Biblos-e Archivo. Repositorio Institucional de la UAM
collection Biblos-e Archivo. Repositorio Institucional de la UAM
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869404733147447296
score 15,300724

Similares en LA Referencia