Α-Robust H1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions. (15th July 2022) - An, Na; Zhao, Guoye; Huang, Chaobao; Yu, Xijun
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 : = 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.