In this work, the superdiffusion equation with a Caputo derivative of order α ∈ (1 , 2) is considered. Some priori bounds on certain derivatives of the solution show that the solution exhibits a weak singularity at the initial time t = 0. To resolve this initial singularity, we rewrite the superdiffusion equation as a coupled system by introducing a intermediate variable p : = D t α / 2 (u − t u 1) , and adopt the L1 scheme and Alikhanov scheme on graded meshes in temporal direction. In spatial direction, the conforming finite element method is used. Furthermore, we derive the H 1 -norm stability result. It is worth noting that some priori bounds on certain derivatives of p are obtained, on basis of which, we derive an α -robust prior error estimate with optimal H 1 -norm convergence order. Finally, we provide the numerical experiment to further verify our theoretical analysis. [ABSTRACT FROM AUTHOR]