Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data

This paper establishes the convergence of a time-steeping scheme for time fractional diffusion problems with nonsmooth data. We first analyze the regularity of the model problem with nonsmooth data, and then prove that the time-steeping scheme possesses optimal convergence rates in $L^2(0,T;L^2(\Omega))$-norm and $L^2(0,T;H_0^1(\Omega))$-norm with respect to the regularity of the solution. Finally, numerical results are provided to verify the theoretical results.