Α-Robust H1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions. (15th July 2022)

• Record Type: Journal Article
• Authors: An, Na; Zhao, Guoye; Huang, Chaobao; Yu, Xijun
• Is Part Of: Computers & mathematics with applications. Volume 118(2022)
• Page Start: 159
• Page End: 170
• Publication Date: 2022-07-15
• Subjects: The superdiffusion equation -- L1 scheme -- Alikhanov scheme -- The finite element method -- Weak singularity; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.05.017
• Languages: English
• ISSNs: 0898-1221

Abstract

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.