Numerical Analysis/Optimal Control
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.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.032
Negreanu, Mihaela 1; Zuazua, Enrique 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
@article{CRMATH_2004__338_5_413_0,
     author = {Negreanu, Mihaela and Zuazua, Enrique},
     title = {Convergence of a multigrid method for the controllability of a 1-d wave equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {413--418},
     publisher = {Elsevier},
     volume = {338},
     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2003.11.032},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2003.11.032/}
}
TY  - JOUR
AU  - Negreanu, Mihaela
AU  - Zuazua, Enrique
TI  - Convergence of a multigrid method for the controllability of a 1-d wave equation
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 413
EP  - 418
VL  - 338
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2003.11.032/
DO  - 10.1016/j.crma.2003.11.032
LA  - en
ID  - CRMATH_2004__338_5_413_0
ER  - 
%0 Journal Article
%A Negreanu, Mihaela
%A Zuazua, Enrique
%T Convergence of a multigrid method for the controllability of a 1-d wave equation
%J Comptes Rendus. Mathématique
%D 2004
%P 413-418
%V 338
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2003.11.032/
%R 10.1016/j.crma.2003.11.032
%G en
%F CRMATH_2004__338_5_413_0
Negreanu, Mihaela; Zuazua, Enrique. 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. http://www.numdam.org/articles/10.1016/j.crma.2003.11.032/

[1] Asch, M.; Lebeau, G. Geometrical aspects of exact boundary controllability for the wave equation – a numerical study, ESAIM: Control Optim. Calc. Var., Volume 3 (1998), pp. 163-212

[2] Glowinski, R. 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

[3] Infante, J.A.; Zuazua, E. Boundary observability for the space discretization of the one-dimensional wave equation, Math. Model. Numer. Anal., Volume 33 (1999) no. 2, pp. 407-438

[4] Lions, J.-L. Contrôlabilité exacte, stabilisation et perturbations du systemes distribués. Tome 1. Contrôlabilité exacte, RMA, vol. 8, Masson, 1988

[5] Micu, S. Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math., Volume 91 (2002) no. 4, pp. 723-768

[6] E. Zuazua, Propagation, observation, control and numerical approximation of waves, Preprint, 2003

Cited by Sources: