Afrika Matematika

, Volume 29, Issue 3–4, pp 557–574 | Cite as

Uniform interior stabilization for the finite difference fully-discretization of the 1-D wave equation



In this paper, we are interested to prove the uniform exponential decay of the energy for the finite difference fully-discretization of 1-D wave equation with an interior damping at \(\xi \). To this end, we decompose the fully discrete system into two subsystems: a conservative system, and a non conservative one. We show under a numerical hypothesis on \(\xi \) that a uniform observability inequality holds for a conservative system, when the mesh size tends to zero using Fourier series technique. Then, we use the observability inequality to prove the uniform exponential decay of the energy for the damped system. Finally, we describe some numerical experiments to illustrate the exponential decay of the energy.


Interior stabilization Pointwise internal damping Uniform exponential decay Uniform interior observability Finite difference method Fully-discretization 

Mathematics Subject Classification

93D15 93B07 65N06 65N22 


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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.National School of Applied SciencesIbn Zohr UniversityAgadirMorocco

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