Motivée par deux problématiques liées aux plasmas de tokamak, cette thèse propose deux méthodes d'approximation numérique pour deux problèmes mathématiques s'y rattachant. D'une part, pour l'étude du transport turbulent de particules, une méthode numérique basée sur les schémas de subdivision est présentée pour la simulation de trajectoires de particules dans un champ de vitesse fortement variable. D'autre part, dans le cadre de la modélisation de l'interaction plasma-paroi, une méthode de pénalisation est proposée pour la prise en compte de conditions aux limites de type Neumann ou Robin. Analysées sur des problèmes modèles de complexité croissante, ces méthodes sont enfin appliquées dans des situations plus réalistes d'intérêt pratique dans l'étude du plasma de bord. / Motivated by two issues related to tokamak plasmas, this thesis proposes two numerical approximation methods for two mathematical problems associated with them. On the one hand, in order to study the turbulent transport of particles, a numerical method based on subdivision schemes is presented for the simulation of particle trajectories in a strongly varying velocity field. On the other hand, in the context of modeling the plasma-wall interaction, a penalization method is proposed to take into account Neumann or Robin boundary conditions. Analyzed on model problems of increasing complexity, these methods are finally applied in more realistic situations of practical interest in the study of the edge plasma.

In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished