Formula omitted]-robust [formula omitted]-norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation

Abstract

The sharp H1-norm error estimate of a finite difference method for two-dimensional time-fractional diffusion equation is established, where the Caputo time-fractional derivative term is approximated by Alikhanov scheme on graded mesh. Under reasonable assumption that the exact solution has typical weak singularity at initial time, the temporal convergence order of the fully scheme is O(N−min{rα,2}) and the error bound does not blow up when α approximates 1−. The theoretical analysis utilizes an improved discrete fractional Grönwall inequality, and the correctness of the theoretical result is verified by numerical experiments.