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dc.contributor.authorMarica, A.
dc.contributor.authorZuazua, E.
dc.date.accessioned2017-02-21T08:18:19Z
dc.date.available2017-02-21T08:18:19Z
dc.date.issued2012-12-31
dc.identifier.issn0036-1429
dc.identifier.urihttp://hdl.handle.net/20.500.11824/559
dc.description.abstractWe 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.
dc.formatapplication/pdf
dc.language.isoengen_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/es/en_US
dc.subjectAcoustic/optic mode
dc.subjectBi-grid algorithm
dc.subjectFourier truncation method
dc.subjectObservability/controllability property
dc.subjectQuadratic finite element method
dc.subjectUniform mesh
dc.subjectVanishing group velocity
dc.titleOn the quadratic finite element approximation of one-dimensional waves: Propagation, observation, and control
dc.typeinfo:eu-repo/semantics/articleen_US
dc.identifier.doi10.1137/110839503
dc.relation.publisherversionhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-84868350564&doi=10.1137%2f110839503&partnerID=40&md5=219a69f3944c9d2a4c2bf3a427a62733
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessen_US
dc.type.hasVersioninfo:eu-repo/semantics/publishedVersionen_US
dc.journal.titleSIAM Journal on Numerical Analysisen_US


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Reconocimiento-NoComercial-CompartirIgual 3.0 España
Except where otherwise noted, this item's license is described as Reconocimiento-NoComercial-CompartirIgual 3.0 España