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

TL;DR

This paper introduces efficient multiscale finite element methods with time-splitting for solving semiclassical nonlinear Schrödinger equations with random potentials, achieving high accuracy and reduced computational cost.

Contribution

The paper develops novel multiscale finite element methods with rigorous error analysis and a reduced basis approach for the semiclassical NLSE with randomness, enhancing efficiency and accuracy.

Findings

Second-order accuracy in space and time

Almost first-order convergence in random space

Validated effectiveness through 1D and 2D numerical examples

Abstract

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 the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the $L^2$ 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 present a multiscale reduced basis 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.