Couplages FEM-BEM faibles et optimisés pour des problèmes de diffraction harmoniques en acoustique et en électromagnétisme

In this doctoral dissertation, we propose new methods for solving acoustic and electromagnetic three-dimensional harmonic scattering problems for which the scatterer is penetrable and inhomogeneous. The resolution of such problems is key in the computation of sonar and radar cross sections (SCS and RCS). However, this task is known to be difficult because it requires discretizing partial differential equations set in an exterior domain. Being unbounded, this domain cannot be meshed thus hindering a volume finite element resolution. There are two standard approaches to overcome this difficulty. The first one consists in truncating the exterior domain and renders possible a volume finite element resolution. Given that they approximate the scattering problems at the continuous level, truncation methods may however not be accurate enough for SCS and RCS computations. Inhomogeneous penetrable harmonic scattering problems can also be solved by coupling a volume variational formulation associated with the scatterer and surface integral equations related to the exterior domain. This approach is known as FEM-BEM coupling (Finite Element Method-Boundary Element Method). It is of great interest because it is exact at the continuous level. Classical FEM-BEM couplings are qualified as strong because they couple the volume variational formulation and the surface integral equations within one unique formulation. They are however not suited for solving high-frequency problems. To remedy this drawback, other FEM-BEM couplings, said to be weak, have been proposed. These couplings are actually domain decomposition algorithms iterating between the scatterer and the exterior domain. In this thesis, we introduce new acoustic and electromagnetic weak FEM-BEM couplings based on recently developed Padé approximations of Dirichlet-to-Neumann and Magnetic-to-Electric operators. The number of iterations required to solve these couplings is only slightly dependent on the frequency and the mesh refinement. The weak FEM-BEM couplings that we propose are therefore suited to accurate SCS and RCS computations at high frequencies.