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Issue ESAIM: Proc.
Volume 18, 2007
Paris-sud working group on modelling and scientific computing 2006-2007
Page(s) 23 - 37
DOI http://dx.doi.org/10.1051/proc:071803
Published online 12 September 2007

ESAIM: Proc., 2007, Vol. 18, pp. 23-37
DOI: 10.1051/proc:071803

Décomposition de Helmholtz par ondelettes : convergence d'un algorithme itératif

Erwan Deriaz1, Kai Bittner2 and Valérie Perrier1

1  Laboratoire de Modélisation et Calcul de l'IMAG, BP 53, 38 041 Grenoble cedex 9, France; ;
2  Université d'Ulm, Helmholtzstr. 18, D-89069 Ulm, Allemagne;

erwan.deriaz@imag.fr
valerie.perrier@imag.fr
kai.bittner@uni-ulm.de

(Published online: 12 September 2007)

Abstract
In what follows, we present tensor-product divergence-free and curl-free wavelets, and we define associated projectors. These projectors permit the construction of an iterative algorithm for the computation of the Helmholtz decomposition in terms of wavelets. This Helmholtz decomposition is localized in space, in contrast to a Helmholtz decomposition calculated by the Fourier transform. Finally, we show the convergence of the algorithm in 2 and 3 dimensions for the particular case of Shannon wavelets.


Résumé
Dans ce qui suit, on présente des ondelettes à divergence nulle et des ondelettes à rotationnel nul, et on définit des projecteurs associés : ces projecteurs permettent de construire un algorithme itératif de décomposition de Helmholtz par ondelettes. Cette décomposition de Helmholtz est localisée en espace, contrairement à une décomposition de Helmholtz qui serait calculée par transformée de Fourier. On démontre la convergence de l'algorithme en dimensions 2 et 3, dans le cas particulier des ondelettes de Shannon.


Mathematics Subject Classification. 42C40, 65T60, 76F65

Key words: ondelettes, décomposition de Helmholtz, champs incompressibles


© EDP Sciences, ESAIM 2007


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