A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

Abstract

In this paper, a class of nonlinear Riesz space-fractional Schrodinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis.