On The Eigenvalues of the Spectral Second Order Differentiation Operator and Application to the Boundary Observability of the Wave Equation

Authors:
T. Z. Boulmezaoud;J. M. Urquiza
Affiliations:
Laboratoire de Mathématiques de Versailles, Université de Versailles SQY, Versailles, France 78035;Centre de Recherches Mathématiques, Université de Montréal, Montréal (Québec), Canada H3C 3J7
Venue:
Journal of Scientific Computing
Year:
2007

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Abstract

The behaviour of the eigenvalues of the spectral second-order differentiation operator is studied and the results are used to investigate the boundary observability of the one dimensional wave equation approximated with a spectral Galerkin method. New explicit estimates of the discrete eigenvalues are given. These estimates improve the previous results on the subject especially for the portion of eigenvalues converging exponentially to those of the continuous problem. Although the boundary observability property of the discretized wave equation is not uniform with respect to the discretization parameter, we show that a uniform observability estimate can be obtained by filtering out the highest eigenmodes.