On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control

We study the propagation, observation, and control properties of the quadratic P 2-classical finite element semidiscretization of the one-dimensional wave equation on a bounded interval. A careful Fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum: the acou...

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Authors: Marica, A., Zuazua, E.
Format: article
Status:Published version
Publication Date:2012
Country:España
Institution:Basque Center for Applied Mathematics (BCAM)
Repository:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/559
Online Access:http://hdl.handle.net/20.500.11824/559
Access Level:Open access
Keyword:Acoustic/optic mode
Bi-grid algorithm
Fourier truncation method
Observability/controllability property
Quadratic finite element method
Uniform mesh
Vanishing group velocity
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spelling On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and controlMarica, A.Zuazua, E.Acoustic/optic modeBi-grid algorithmFourier truncation methodObservability/controllability propertyQuadratic finite element methodUniform meshVanishing group velocityWe study the propagation, observation, and control properties of the quadratic P 2-classical finite element semidiscretization of the one-dimensional wave equation on a bounded interval. A careful Fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum: the acoustic one, of physical nature, and the optic one, related to the perturbations that this second-order finite element approximation introduces with respect to the P 1 one. On both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero. This shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme. As a consequence of this, the controls of the discrete waves may blow up as the mesh size tends to zero. To remedy these high-frequency pathologies, we design filtering mechanisms based on the Fourier truncation method or on a bi-grid algorithm, for which one can recover the uniformity of the observability constant in a finite time and, consequently, the possibility to control with uniformly bounded L 2-controls appropriate projections of the solutions. This also allow us to show that, by relaxing the control requirement, the controls are uniformly bounded and converge to the continuous ones as the mesh size tends to zero.201720172012info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/559reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://www.scopus.com/inward/record.uri?eid=2-s2.0-84868350564&doi=10.1137%2f110839503&partnerID=40&md5=219a69f3944c9d2a4c2bf3a427a62733Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/5592026-06-19T12:47:47Z
dc.title.none.fl_str_mv On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
title On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
spellingShingle On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
Marica, A.
Acoustic/optic mode
Bi-grid algorithm
Fourier truncation method
Observability/controllability property
Quadratic finite element method
Uniform mesh
Vanishing group velocity
title_short On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
title_full On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
title_fullStr On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
title_full_unstemmed On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
title_sort On the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
dc.creator.none.fl_str_mv Marica, A.
Zuazua, E.
author Marica, A.
author_facet Marica, A.
Zuazua, E.
author_role author
author2 Zuazua, E.
author2_role author
dc.subject.none.fl_str_mv Acoustic/optic mode
Bi-grid algorithm
Fourier truncation method
Observability/controllability property
Quadratic finite element method
Uniform mesh
Vanishing group velocity
topic Acoustic/optic mode
Bi-grid algorithm
Fourier truncation method
Observability/controllability property
Quadratic finite element method
Uniform mesh
Vanishing group velocity
description We study the propagation, observation, and control properties of the quadratic P 2-classical finite element semidiscretization of the one-dimensional wave equation on a bounded interval. A careful Fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum: the acoustic one, of physical nature, and the optic one, related to the perturbations that this second-order finite element approximation introduces with respect to the P 1 one. On both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero. This shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme. As a consequence of this, the controls of the discrete waves may blow up as the mesh size tends to zero. To remedy these high-frequency pathologies, we design filtering mechanisms based on the Fourier truncation method or on a bi-grid algorithm, for which one can recover the uniformity of the observability constant in a finite time and, consequently, the possibility to control with uniformly bounded L 2-controls appropriate projections of the solutions. This also allow us to show that, by relaxing the control requirement, the controls are uniformly bounded and converge to the continuous ones as the mesh size tends to zero.
publishDate 2012
dc.date.none.fl_str_mv 2012
2017
2017
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/559
url http://hdl.handle.net/20.500.11824/559
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://www.scopus.com/inward/record.uri?eid=2-s2.0-84868350564&doi=10.1137%2f110839503&partnerID=40&md5=219a69f3944c9d2a4c2bf3a427a62733
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
repository.name.fl_str_mv
repository.mail.fl_str_mv
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