Efficient finite element methods for semiclassical nonlinear Schr\"odinger equations with random potentials

In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schr\"odinger equation (NLSE) with random potentials. We then introduce a multiscale method to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze the corresponding time-splitting multiscale reduced methods} for the semiclassical NLSE with random potentials. We provide the L² error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we introduce the proper orthogonal decomposition method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials.