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

Farah Abdallah; Serge Nicaise; Julie Valein; Ali Wehbe

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 3, page 844-887
- ISSN: 1292-8119

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topAbdallah, Farah, et al. "Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 844-887. <http://eudml.org/doc/272848>.

@article{Abdallah2013,

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.},

author = {Abdallah, Farah, Nicaise, Serge, Valein, Julie, Wehbe, Ali},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {stability; wave equation; numerical approximations; numerical viscosity term; second-order evolution equations; convergence},

language = {eng},

number = {3},

pages = {844-887},

publisher = {EDP-Sciences},

title = {Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications},

url = {http://eudml.org/doc/272848},

volume = {19},

year = {2013},

}

TY - JOUR

AU - Abdallah, Farah

AU - Nicaise, Serge

AU - Valein, Julie

AU - Wehbe, Ali

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2013

PB - EDP-Sciences

VL - 19

IS - 3

SP - 844

EP - 887

AB - 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.

LA - eng

KW - stability; wave equation; numerical approximations; numerical viscosity term; second-order evolution equations; convergence

UR - http://eudml.org/doc/272848

ER -

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