Couplage par décomposition de domaine optimisée de formulations intégrales et éléments finis d’ordre élevé pour l’électromagnétisme

In terms of computational methods, solving three-dimensional time-harmonic electromagnetic scattering problems is known to be a challenging task, most particularly in the high frequency regime and for dielectric and inhomogeneous scatterers. Indeed, it requires to discretize a system of partial differential equations set in an unbounded domain. In addition, considering a small wavelength λ in this case, naturally requires very fine meshes, and therefore leads to very large number of degrees of freedom. A standard approach consists in
combining integral equations for the exterior domain and a weak formulation for the interior
domain (the scatterer) resulting in a formulation coupling the Boundary Element Method (BEM) and the Finite Element Method (FEM). Although natural, this approach has some major drawbacks. First, this standard coupling method yields a very large system having a matrix with sparse and dense blocks, which is therefore generally hard to solve and not directly adapted to compression methods. Moreover, it is not possible to easily combine two
pre-existing solvers, one FEM solver for the interior domain and one BEM solver for the exterior domain, to construct a global solver for the original problem.
In this thesis, we present a well-conditioned weak coupling formulation between the boundary element method and the high-order finite element method, allowing the construction of such a solver. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Padé
approximants of the nonlocal Magnetic-to-Electric operators. The number of iterations required to solve this weak coupling is only slightly dependent on the geometry configuration, the frequency, the contrast between the subdomains and the mesh refinement.