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Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

Published online by Cambridge University Press:  03 June 2013

Farah Abdallah ,
Serge Nicaise ,
Julie Valein  and
Ali Wehbe
Farah Abdallah
Affiliation:
Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France; Farah.Abdallah@meletu.univ-valenciennes.fr
Serge Nicaise
Affiliation:
Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr
Julie Valein
Affiliation:
Institut Elie Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-lès-Nancy Cedex France; Julie.Valein@iecn.u-nancy.fr
Ali Wehbe
Affiliation:
Université Libanaise, Ecole Doctorale des Sciences et de Technologie, Hadath, Beyrouth, Liban; ali.wehbe@ul.edu.lb
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Abstract

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter–Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

Keywords

Stabilitywave equationnumerical approximations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 844 - 887
Copyright
© EDP Sciences, SMAI, 2013

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