α-Robust H1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions

Abstract

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:=Dtα/2(u−tu1), 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 H1-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 H1-norm convergence order. Finally, we provide the numerical experiment to further verify our theoretical analysis.