Modified L1 Crank–Nicolson finite element methods with unconditional convergence for nonlinear time-fractional Schrödinger equations

Abstract

The paper focuses on numerically solving the nonlinear time fractional Schrödinger equations. The modified $L1$ Crank–Nicolson scheme is used for the time discretization and the Galerkin finite element approximation is used in the spatial direction. Besides, we provide the proofs of stability by mathematical induction and unconditionally optimal error estimates by a discrete fractional Grönwall inequality derived in this paper, Sobolev embedding theorems and some other inequalities for the fully discrete linearized modified $L1$ Galerkin finite element scheme. Furthermore, a transformed $L1$ Crank–Nicolson Galerkin finite element method based on the change of variable $t=s^{1/\alpha }$ is constructed on the uniform space–time mesh for solving the nonlinear time fractional Schrödinger equations with nonsmooth solutions. Finally, the numerical experiments clearly and accurately demonstrate the rationality of the numerical scheme and the correctness of the theoretical results.