A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations

Abstract

We consider the numerical simulation of the coupled nonlinear space fractional Schrodinger equations. Based on the Galerkin finite element method in space and the Crank-Nicolson (CN) difference method in time, a fully discrete scheme is constructed. Firstly, we focus on a rigorous analysis of conservation laws for the discrete system. The definitions of discrete mass and energy here correspond with the original ones in physics. Then, we prove that the fully discrete system is uniquely solvable. Moreover, we consider the unconditionally convergent properties (that is to say, we complete the error estimates without any mesh ratio restriction). We derive $L^{2}$ -norm error estimates for the nonlinear equations and $L^{\infty}$ -norm error estimates for the linear equations. Finally, some numerical experiments are included showing results in agreement with the theoretical predictions.