Méthode de décomposition de domaines pour l'équation de Schrödinger. (Domain decomposition method for Schrödinger equation)

This thesis focuses on the development and the implementation of domain decomposition methods for the linear and non-linear, one dimensional and two dimensional Schrödinger equations. In the first part, we focus on the Schwarz waveform relaxation method (SWR) for the one dimensional Schrödinger equation. In the case the potential is linear and time-independent, we propose a new algorithm that is scalable and allows a significant reduction of computation time compared with the classical algorithm. For a time-dependent linear potential or a non-linear potential, we use a previously defined linear operator as preconditioner. This ensures high scalability. We also generalize the work of Halpern and Szeftel on transmission condition. We use the absorbing boundary conditions recently constructed by Antoine, Besse and Klein as the transmission condition. We also carry the codes developed on Cpu on Gpus accelerator (graphics card) for the one dimensional Schrödinger equation. The second part concerns the SWR method and the domain decomposition in space method for the Schrödinger equation in two dimensions. We generalize the new algorithm and the preconditioned algorithm proposed in the first part to the case of two dimensions. Furthermore, in Chapter 6, we generalize the work of Loisel on the optimized Schwarz method with cross points for the Laplace equation, which leads to the SWR method with cross points. In the last part, we apply the domain decomposition methods présented in the previous chapters to the simulation of Bose-Einstein condensate, that could not only reduce the total computation time, but also realise the simulations which are not possible on a single calculation node.