Contributions to the numerical study of networks in electromagnetism and of the boundary layer in fluid mechanics.

Summary The first part of the thesis is devoted to the study of the reflection of electromagnetic waves by periodical structures arranged on various tones. We are first interested in the case where the wavelength of electromagnetic phenomena is of the same order as the period of the structure. We consider the problem from the point of view of shape optimization. Concretely, we try to optimize the interface between two dielectric layers in a polar cell, in order to increase its efficiency. This study is mainly numerical, and we consider the original problem and the relaxed problem, where this time we try to optimize a mixing coefficient. We are then interested in the case where the wavelength is greater than the size of the period. We can then use asymptotic expansion techniques to find an equivalent boundary condition, i. e. The second part of the paper is devoted to the analysis of the problem of the period structure. The second part is devoted to the numerical study of the boundary layer for incompressible viscous fluid flows with large reynolds numbers. Finally, we consider a mixed finite element method where the current function, which does not develop boundary layers, is discretized on a much coarser grid than the vorticity which has fast variations.