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

Published online by Cambridge University Press:  19 December 2008

Sylvain Ervedoza
Sylvain Ervedoza*
Affiliation:
Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France. sylvain.ervedoza@math.uvsq.fr
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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.

Keywords

Spectrumobservabilitywave equationsemi-discrete systemscontrollabilitystabilization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 298 - 326
Copyright
© EDP Sciences, SMAI, 2008

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