Domain decomposition algorithms for two dimensional linear Schr\"odinger equation

This paper deals with two domain decomposition methods for two dimensional linear Schr{\"o}dinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the free Schr{\"o}dinger equation and a preconditioned algorithm for the general Schr{\"o}dinger equation. These algorithms are studied numerically, which shows that the two new algorithms could accelerate the convergence and reduce the computation time. Besides the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.