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We provide a rigorous asymptotic method to build approximate conditions equivalent to a thin layer. With these conditions the electromagnetic field may be computed in domains such as a biological cell, without meshing the membrane. The influence of the membrane is replaced by an appropriate condition on the inner domain, while in the thin layer, the approximate field is explicitly known. We give error estimates, which validate our asymptotic method, and we present a few numerical results performed with the finite-element method. Index Terms--Asymptotic methods, finite-element method, thin layers.

The numerical solution of the three-dimensional time-harmonic Maxwell equations using high order methods such as discontinuous Galerkin formulations require efficient solvers. A domain decomposition strategy is introduced for this purpose. This strategy is based on optimized Schwarz methods applied to the first order form of the Maxwell system and leads to the best possible convergence of these algorithms. The principles are explained for a 2D model problem and numerical simulations confirm the predicted theoretical behavior. The efficiency is further demonstrated on more realistic 3D geometries including a bioelectromagnetism application. Index Terms--Discontinuous Galerkin methods, domain decomposition methods, optimized interface conditions.

An accurate method to compute dual field in high frequency time harmonic problem is presented. From a primal field obtained by a vector finite element discretization, the dual field is obtained without numerical derivation by using a least square argument. The accuracy of the method is compared with the natural method using shape function derivatives. Index Terms--Edge elements, finite element method, post-processing.

An efficient preconditioner for systems issued from the finite element discretization of time harmonic Maxwell's equations with absorbing boundary conditions is presented. It is based on the Helmholtz decomposition of the electromagnetic field and its discrete counterpart. It is compared to a classical preconditioner on both simple and realistic problems. Its behavior is also evaluated on meshes showing different characteristics. Index Terms--Electromagnetic fields, finite-element methods, numerical analysis, scattering.