Journal of Scientific Computing

, Volume 31, Issue 3, pp 307–345 | Cite as

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

Article

 

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 methods differentiation matrix eigenvalues Lommel polynomials wave equation observability filtering 

2000 Mathematics Subject Classification

65N35 65D05 65F15 93B07 93B60 

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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

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