Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Sylvain Ervedoza

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 2, page 298-326
  • ISSN: 1292-8119

Abstract

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The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.

How to cite

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Ervedoza, Sylvain. "Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 298-326. <http://eudml.org/doc/250780>.

@article{Ervedoza2010,
abstract = { The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author. },
author = {Ervedoza, Sylvain},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Spectrum; observability; wave equation; semi-discrete systems; controllability; stabilization; spectrum},
language = {eng},
month = {4},
number = {2},
pages = {298-326},
publisher = {EDP Sciences},
title = {Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes},
url = {http://eudml.org/doc/250780},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Ervedoza, Sylvain
TI - Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 298
EP - 326
AB - The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.
LA - eng
KW - Spectrum; observability; wave equation; semi-discrete systems; controllability; stabilization; spectrum
UR - http://eudml.org/doc/250780
ER -

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