Thin Layer Models for Electromagnetism

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer.These models appear as first order approximations of the electromagnetic field.They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer,that makes possible to replace the thin layer by approximate conditions.We present the advantages and the drawbacks of several approximations together with numerical validations and simulations.The main motivation of this work concerns the computation of electromagnetic field in biological cells.The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane,which provides a specific behavior of the electromagnetic field at low frequencies.

Matched Asymptotic Expansions of the Eigenvalues of a 3-D Boundary-Value Problem Relative to Two Cavities Linked by a Hole of Small Size

In this article,we consider a domain consisting of two cavities linked by a hole of small size.We derive a numerical method to compute an approximation of the eigenvalues of an elliptic operator without refining in the neighborhood of the hole.Several convergence rates are obtained and illustrated by numerical simulations.

FullApertureReconstruction of theAcoustic Far-Field Pattern from Few Measurements

We propose a numerical procedure to extend to full aperture the acoustic farfield pattern(FFP)when measured in only few observation angles.The reconstruction procedure is a multi-step technique that combines a total variation regularized iterative method with the standard Tikhonov regularized pseudo-inversion.The proposed approach distinguishes itself from existing solution methodologies by using an exact representation of the total variation which is crucial for the stability and robustness of Newton algorithms.We present numerical results in the case of two-dimensional acoustic scattering problems to illustrate the potential of the proposed procedure for reconstructing the full aperture of the FFP from very few noisy data such as backscattering synthetic measurements.

A Modified Discontinuous GalerkinMethod for Solving Efficiently Helmholtz Problems

A new solution methodology is proposed for solving efficiently Helmholtz problems.The proposed method falls in the category of the discontinuous Galerkin methods.However,unlike the existing solution methodologies,this method requires solving(a)well-posed local problems to determine the primal variable,and(b)a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges.Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.