Journal of Scientific Computing

, Volume 31, Issue 3, pp 307–345

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

Article

DOI: 10.1007/s10915-006-9106-8

Cite this article as:
Boulmezaoud, T.Z. & Urquiza, J.M. J Sci Comput (2007) 31: 307. doi:10.1007/s10915-006-9106-8

 

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.

Keywords

Spectral methodsdifferentiation matrixeigenvaluesLommel polynomialswave equationobservabilityfiltering

2000 Mathematics Subject Classification

65N3565D0565F1593B0793B60

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de VersaillesUniversité de Versailles SQYVersaillesFrance
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontréal (Québec)Canada