Convergence of a multigrid method for the controllability of a 1-d wave equation

Mihaela Negreanu; Enrique Zuazua

doi:10.1016/j.crma.2003.11.032

Numerical Analysis/Optimal Control

Convergence of a multigrid method for the controllability of a 1-d wave equation
[Convergence d'une méthode multigrille pour la contrôlabilité d'une équation d'ondes 1-d]

Présenté par : Philippe G. Ciarlet

Mihaela Negreanu ¹ ; Enrique Zuazua ²

¹ Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
² Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 413-418.

Résumé
Abstract

On considère le problème de l'approximation numérique du contrôle frontière de l'équation des ondes. Il est maintenant bien connu que la plupart des méthodes de différences finies et éléments finis classiques ne donnent pas des approximations convergentes à cause des instabilités dues aux hautes fréquences. Plusieurs remèdes on été proposés dans la littérature pour compenser ce fait : régularisation de Tychonoff, filtrage de Fourier, éléments finis mixtes,… Dans cette Note on démontre la convergence de la méthode de multi-grille proposée par Glowinski (J. Comput. Phys. 103 (2) (1992) 189–221) dans le cas de l'approximation semi-discrète de l'équation des ondes par différences finies.

We consider the problem of computing numerically the boundary control for the wave equation. It is by now well known that, due to high frequency spurious oscillations, numerical instabilities occur and may led to the failure of convergence of some apparently natural numerical algorithms. Several remedies have been proposed in the literature to compensate this fact: Tychonoff regularization, Fourier filtering, mixed finite elements,… In this Note we prove that the two-grid method proposed by Glowinski (J. Comput. Phys. 103 (2) (1992) 189–221) does indeed provide a convergent algorithm. This is done in the context of the finite-difference semi-discrete approximation of the 1-d wave equation.

Reçu le : 2003-11-23
Accepté le : 2003-11-26
Publié le : 2004-01-29

DOI : 10.1016/j.crma.2003.11.032

Affiliations des auteurs :

Mihaela Negreanu ¹ ; Enrique Zuazua ²

¹ Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
² Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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     title = {Convergence of a multigrid method for the controllability of a 1-d wave equation},
     journal = {Comptes Rendus. Math\'ematique},
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     language = {en},
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Mihaela Negreanu; Enrique Zuazua. Convergence of a multigrid method for the controllability of a 1-d wave equation. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 413-418. doi : 10.1016/j.crma.2003.11.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.11.032/

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