Local convergence analysis of L1-ADI scheme for two-dimensional reaction-subdiffusion equation

In this paper, we propose an alternating direction implicit (ADI) difference method to solve the two-dimensional time-fractional reaction-subdiffusion equation with weakly singular solutions, where L1 scheme on graded mesh is used to capture the initial weak singularity. Global and local convergences of L1-ADI scheme on graded mesh are studied with the comparison principle. The temporal global convergence rate of the fully discrete L1-ADI scheme is \(O(N^{-\min \{ 2-\alpha ,2\alpha ,\alpha r\}})\) $O(N^{-min\{2-\alpha ,2\alpha ,\alpha r\}})$ and the temporal local convergence rate can attain \(O(N^{-\min \{ 2-\alpha , 2\alpha \}})\) $O(N^{-min\{2-\alpha ,2\alpha \}})$ when \(r> 2-\alpha \) $r>2-\alpha$ , where N, \(\alpha \) $\alpha$ , and r are the time partition number, order of fractional derivative, and grading parameter, respectively. Numerical experiments are proposed to verify the sharpness of our theoretical analysis.