Issue |
ESAIM: COCV
Volume 19, Number 3, July-September 2013
|
|
---|---|---|
Page(s) | 844 - 887 | |
DOI | https://doi.org/10.1051/cocv/2012036 | |
Published online | 03 June 2013 |
Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
1
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
2
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
3
Institut Elie Cartan Nancy (IECN), Nancy-Université &
INRIA (Project-Team CORIDA), 54506
Vandoeuvre-lès-Nancy Cedex
France
Julie.Valein@iecn.u-nancy.fr
4
Université Libanaise, Ecole Doctorale des Sciences et de
Technologie, Hadath, Beyrouth, Liban
ali.wehbe@ul.edu.lb
Received: 21 December 2011
Revised: 7 May 2012
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.
Mathematics Subject Classification: 65M60 / 35L05 / 35L15
Key words: Stability / wave equation / numerical approximations
© EDP Sciences, SMAI, 2013
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