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

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.