Stability results for the approximation of weakly coupled wave equations

Farah Abdallah; Serge Nicaise; Julie Valein; Ali Wehbe

doi:10.1016/j.crma.2011.12.004

Partial Differential Equations

Stability results for the approximation of weakly coupled wave equations

Presented by: Gilles Lebeau

Farah Abdallah ^1,²; Serge Nicaise ¹; Julie Valein ³; Ali Wehbe ²

¹ Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, institut des sciences et techniques, 59313 Valenciennes cedex 9, France
² Université Libanaise, faculté des sciences 1 & Hadath, Beyrouth, Liban
³ Institut Elie-Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 29-34.

Abstract
Résumé

In this Note, we consider the approximation of two coupled wave equations with internal damping. 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 (since the spectrum of the spatial operator associated with the undamped system satisfies the generalized gap condition).

Dans cette Note, nous considérons lʼapproximation de deux équations des ondes couplées avec dissipation interne. Notre but est dʼamortir les modes étranges en introduisant des termes de viscosité numérique. Avec ces termes de viscosité, nous montrons la décroissance exponentielle ou polynomiale du schéma discret lorsque le problème continu a une telle décroissance (puisque le spectre de lʼopérateur spatial associé au système sans dissipation satisfait la condition du gap généralisé).

Received: 2011-06-07
Accepted: 2011-12-05
Published online: 2012-01-01

DOI: 10.1016/j.crma.2011.12.004

Author's affiliations:

Farah Abdallah ^{1, 2}; Serge Nicaise ¹; Julie Valein ³; Ali Wehbe ²

¹ Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, institut des sciences et techniques, 59313 Valenciennes cedex 9, France
² Université Libanaise, faculté des sciences 1 & Hadath, Beyrouth, Liban
³ Institut Elie-Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France

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     author = {Farah Abdallah and Serge Nicaise and Julie Valein and Ali Wehbe},
     title = {Stability results for the approximation of weakly coupled wave equations},
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     doi = {10.1016/j.crma.2011.12.004},
     language = {en},
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Farah Abdallah; Serge Nicaise; Julie Valein; Ali Wehbe. Stability results for the approximation of weakly coupled wave equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 29-34. doi : 10.1016/j.crma.2011.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.004/

References
Cited by

[1] F. Abdallah, S. Nicaise, J. Valein, A. Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications. Technical report, 2011, submitted for publication.

[2] F. Alabau; P. Cannarsa; V. Komornik Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., Volume 2 (2002) no. 2, pp. 127-150

[3] K. Ammari; M. Tucsnak Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., Volume 6 (2001), pp. 361-386

[4] H.T. Banks; K. Ito; C. Wang Exponentially stable approximations of weakly damped wave equations, Estimation and Control of Distributed Parameter Systems (Vorau, 1990), Internat. Ser. Numer. Math., vol. 100, Birkhäuser, Basel, 1991, pp. 1-33

[5] A. Borichev; Y. Tomilov Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478

[6] P.G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

[7] R. Glowinski Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., Volume 103 (1992) no. 2, pp. 189-221

[8] R. Glowinski; W. Kinton; M.F. Wheeler A mixed finite element formulation for the boundary controllability of the wave equation, Internat. J. Numer. Methods Engrg., Volume 27 (1989) no. 3, pp. 623-635

[9] R. Glowinski; C.H. Li; J.-L. Lions A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods, Japan J. Appl. Math., Volume 7 (1990) no. 1, pp. 1-76

[10] J.A. Infante; E. Zuazua Boundary observability for the space semi-discretizations of the one-dimensional wave equation, M2AN, Volume 33 (1999), pp. 407-438

[11] K. Ito; F. Kappel The Trotter–Kato theorem and approximation of PDEs, Math. Comp., Volume 67 (1998) no. 221, pp. 21-44

[12] S. Nicaise; J. Valein Stabilization of second order evolution equations with unbounded feedback with delay, Control Optim. Calc. Var., Volume 16 (2010), pp. 420-456

[13] K. Ramdani; T. Takahashi; M. Tucsnak Uniformly exponentially stable approximations for a class of second order evolution equations—application to LQR problems, ESAIM Control Optim. Calc. Var., Volume 13 (2007) no. 3, pp. 503-527

[14] L.R. Tcheugoué Tébou; E. Zuazua Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math., Volume 95 (2003) no. 3, pp. 563-598

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