Domain decomposition algorithms for the two dimensional nonlinear Schr{\"o}dinger equation and simulation of Bose-Einstein condensates

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schr{"o}dinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.