# Séminaire POEMS sur les courants de Foucault et l'approximation basse fréquence

We will prove the complete low-frequency asymptotics for time-harmonic Maxwell equations in exterior domains. Starting with introducing the solution theory for time-harmonic electro-magnetic scattering problems via a generalized Fredholm alternative using the limiting absorption principle, we continue with proving an adequate corresponding electro-magneto static solution theory providing also special so-called towers of static solutions. In both cases we will work in polynomially weighted Sobolev spaces. Then a comparison with the whole space solution shows that a generalized asymptotic Neumann series gives the desired asymptotics for low frequencies up to a finite sum of degenerate operators, which can be described explicitly by strongly growing towers. Finally we compare these time-harmonic Maxwell radiation solutions with the corresponding solutions provided by the eddy-current model for low frequencies.

####15h30: Edouard Demaldent "Simulation of electromagnetic non-destructive testing by the boundary element method. The Eddy Current Model as an Asymptotic Form of the Maxwell Model"

Electromagnetic testing is widely used for the characterization of a medium as to the detection of defects. In particular, the eddy current non-destructive testing of tubes in steam generators is determinant to diagnose the integrity of heat exchangers in nuclear industry. A valuable support to the mastering of these processes is brought by modeling and the boundary element method (BEM) is an appropriate simulation tool to many inspection configurations. The department of imaging and simulation for the control at CEA LIST is developing a BEM code devoted to these applications, mostly for eddy current testing. Some of these tools are, or will be, integrated into the CIVA software platform, whose target users are experts in non-destructive testing (non numericians). In this talk, we will start with an introduction of the (usual) boundary element method for the low frequency Maxwell transmission problem that requires, in particular, a quasi-Helmholtz decomposition of the divergence-conforming approximation space. We will then present in detail the eddy current model, that is to say the low frequency and high conductive case, as an asymptotic form of the Maxwell BEM model (joint work with Marc Bonnet, CNRS, UMR POems). In the third part we will review and discuss research and development activities around BEM for electromagnetic testing at CEA LIST, such as the use of high-order boundary elements and of specific domain decomposition strategies.