Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when alpha (the order of fractional derivative) approaches 1(-). It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(tau(2 alpha) + tau t(n)(alpha-1)) at time t = t(n); hence the scheme is globally O(tau(alpha)) accurate in temporal direction, but it is O(tau(min{2 alpha,1)}) when t is away from 0.