In this article, we present the mathematical analysis of the convergence of the linearized Crank-Nicolson Galerkin method for a nonlinear Schrodinger problem related to a domain with a moving boundary. The convergence analysis of the numerical method is carried out for both semi-discrete and fully discrete problems. An optimal error estimate in the $L^2$-norm with order ${O}(τ^2+ h^s),~ 2\leq s\leq r$, where $h$ is the finite element mesh size parameter, $τ$ is the time step, and $r-1$ represents the degree of the finite element polynomial basis. Numerical simulations are provided to confirm the consistency between theoretical and numerical results, validating the method and the order of convergence for different degrees $p\geq 1$ of the Lagrange polynomials and also for Hermite polynomials (degree $p=3$), which form the basis of the approximate solution.
Symplectic Runge-Kutta Semi-discretization for Stochastic Schrödinger Equation
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学/工程计算国家重点实验室）
Low regularity error estimates for high dimensional nonlinear Schrödinger equations
Department of Mathematics, Universit\" a t Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria (L. Ji)；Department of Mathematics, Universit\" a t Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria (A. Ostermann)
Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；National University of Defense Technology（国防科技大学）
