Unconditionally stable finite element method for the variable-order fractional Schrödinger equation with Mittag-Leffler kernel

The Schrödinger equation with variable-order fractional operator is a challenging problem to be solved numerically. In this study, an implicit fully discrete continuous Galerkin finite element method is developed to tackle this equation while the fractional operator is expressed with a nonsingular Mittag-Leffler kernel. To begin with, the finite difference scheme known as the L1 formula is employed to discretize the temporal term. Next, the continuous Galerkin method is used for spatial discretization. This combination ensures accuracy and stability of the numerical approximation. Our next step is to conduct a stability and error analysis of the proposed scheme. Finally, some numerical results are carried out to validate the theoretical analysis.