Volume 19, Number 3, July-September 2013
|Page(s)||844 - 887|
|Published online||03 June 2013|
Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS
2956, Institut des Sciences et Techniques of Valenciennes,
Valenciennes Cedex 9,
2 Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France
3 Institut Elie Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-lès-Nancy Cedex France
4 Université Libanaise, Ecole Doctorale des Sciences et de Technologie, Hadath, Beyrouth, Liban
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.