Ultra-Weak Variational Formulation and efficient Integral Representation in Electromagnetism: a thorough study of the algorithm complexity

Free access article

Issue		ESAIM: PROC Volume 22, 2008 CANUM 2006 - Congrès National d'Analyse Numérique


Page(s)		190 - 197
DOI		10.1051/proc:072224
Published online		01 February 2008

ESAIM: Proc., 2008, Vol. 22, pp. 190-197
DOI: 10.1051/proc:072224

Ultra-Weak Variational Formulation and efficient Integral Representation in Electromagnetism: a thorough study of the algorithm complexity

E. Darrigrand

IRMAR - Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

(... / Published online: 1 February 2008)

Abstract
Many different methods have been developed for the solution of the time-harmonic Maxwell equations in exterior domains at high frequency. Volume based methods have the drawback of needing an artificial boundary far from the obstacle. Integral formulations enable one to avoid this difficulty by solving a problem on the surface of the obstacle, but imply dense systems with bad condition numbers. In this paper we study a coupling of the Ultra-Weak Variational Formulation (UWVF), a volume based method using plane wave basis functions, and an integral representation of the unknown field to obtain an exact artificial boundary condition. More precisely, we detail the study of the complexity of the new algorithm considering a 1-level or multilevel Fast Multipole Method.

Résumé
La résolution des équations de Maxwell en régime harmonique en domaine extérieur, à haute fréquence, a induit de nombreuses méthodes dont les méthodes volumiques nécessitant la considération d'une frontière artificielle pour délimiter le domaine extérieur, ainsi que les formulations intégrales évitant la considération du domaine extérieur mais impliquant des systèmes pleins généralement mal conditionnés. Dans cet article, nous étudions en détail les gains algorithmiques apportés par le couplage d'une méthode de résolution volumique, la formulation variationnelle ultra-faible, et d'une représentation intégrale, via l'utilisation de méthodes multipôles rapides.

Mathematics Subject Classification. ???, ???

Key words: Ultra-Weak Variational Formulation, ABC, integral representation, Fast Multipole Method, algorithm complexity

© EDP Sciences, ESAIM 2007

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