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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. Resume. La resolution desde Maxwell en regime harmonique en domaine exterieur, ` haute frequence, a induit de nombreuses methodes dont les methodes volumiques necessitant la consideration d'une frontiere artificielle pour delimiter le domaine exterieur, ainsi que les formulations integrales ´ la consideration du domaine exterieur mais impliquant des systemes pleins generalement mal conditionnes. Dans cet article, nousetudions en detail les gains algorithmiques apportes par le couplage d'une methode de resolution volumique, la formulation variationnelle ultra-faible, et d'une representation integrale, via l'utilisation de methodes multipoles rapides.

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