Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

Advances in Difference Equations

article

An, Na¹ 

[1] School of Mathematics and Statistics, Shandong Normal University

关键词: Time-fractional diffusion;    Caputo derivative;    Finite element method;    Superconvergence;   

DOI  :  10.1186/s13662-020-02976-4

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time$t=0$ . Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as$r\geq (2-\alpha )/\alpha $ . Numerical results confirm that our analysis is sharp.

【 授权许可】

CC BY