Sur le calcul numérique des modes non-linéaires

Nos travaux portent sur le calcul numérique de modes non linéaires. L'approche adoptée consiste à résoudre par différence finie l'équation différentielle décrivant un mode, s'exprimant comme un problème de transport dont on recherche les conditions initiales donnant des solutions périodiques. Les algorithmes de résolution et d'optimisation sont testés sur un système à deux degrés de liberté et à non linéarités cubiques. Cet exemple nous permet de discuter de la convergence des algorithmes et des problèmes d'implémentation. Les résultats sont également comparés à des calculs par continuation

On the Spectrum of Volume Integral Operators in Acoustic Scattering

Volume integral equations have been used as a theoretical tool in scattering theory for a long time. A classical application is an existence proof for the scattering problem based on the theory of Fredholm integral equations. This approach is described for acoustic and electromagnetic scattering in the books by Colton and Kress [CoKr83, CoKr98] where volume integral equations appear under the name "Lippmann-Schwinger equations". In electromagnetic scattering by penetrable objects, the volume integral equation (VIE) method has also been used for numerical computations. In particular the class of discretization methods known as "discrete dipole approximation" [PuPe73, DrFl94] has become a standard tool in computational optics applied to atmospheric sciences, astrophysics and recently to nano-science under the keyword "optical tweezers", see the survey article [YuHo07] and the literature quoted there. In sharp contrast to the abundance of articles by physicists describing and analyzing applications of the VIE method, the mathematical literature on the subject consists only of a few articles. An early spectral analysis of a VIE for magnetic problems was given in [FrPa84], and more recently [Ki07, KiLe09] have found sufficient conditions for well-posedness of the VIE in electromagnetic and acoustic scattering with variable coefficients. In [CoDK10, CoDS12], we investigated the essential spectrum of the VIE in electromagnetic scattering under general conditions on the complex-valued coefficients, finding necessary and sufficient conditions for well-posedness in the sense of Fredholm in the physically relevant energy spaces. A detailed presentation of these results can be found in the thesis [Sa14]. Publications based on the thesis are in preparation. Curiously, whereas the study of VIE in electromagnetic scattering has thus been completed as far as questions of Fredholm properties are concerned, the simpler case of acoustic scattering does not seem to have been covered in the same depth. It is the purpose of the present paper to close this gap

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

International audienceWe consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay

NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT

In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equation in a rigid duct. This equation models the acoustic propagation in the presence of a flow [1]. We propose a regularized variational formulation of the problem, in the subsonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions.