Error Analysis of an Alternating Direction Implicit Difference Method for 2D Subdiffusion Equation with Initial Singularity

The alternating direction implicit (ADI) scheme is used to numerically solve the 2D subdiffusion equation with initial singularity. The time derivative is defined by the commonly used Caputo fractional derivative, and discretised by the L1 scheme on nonuniform mesh. The finite difference method (FDM) is applied to spatial discretization. The local error analyses of fully discrete scheme under the $L^{2}$-norm and $H^{1}$-norm are strictly established. By selecting the milder grading parameter $r \gt 2-\alpha$, the time convergence rate can reach $O(M^{-\min \{ 2-\alpha, 2\alpha \}})$ in positive time. In order to verify the correctness of the theoretical analysis, some numerical results are presented.