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 Schr¨odinger equation (TFSE). We first decompose the original problem into a coupled system and construct its fully discrete approximation scheme based on the L 2 − 1 σ formula and the finite element method (FEM). We then analyze the stability of the fully discrete scheme. We also rigorously derive its convergence and super-convergence with further regularity assumptions. Finally, we provide several numerical experiments to confirm our theoretical statements.