Superconvergence of FEM combined with the $L2-1_{\sigma}$ scheme for time fractional Schrödinger equations

This paper investigates a numerical solution of the time fractional Schrdinger equation (TFSE).We first decompose the original problem into a coupled system and construct its fully discrete approximation scheme based on the L2 -1 formula and the finite element method (FEM).We then analyze the stability of the fully discrete scheme.We also rigorously derive its convergence and superconvergence with further regularity assumptions.Finally, we provide several numerical experiments to confirm our theoretical statements.