  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.

Numerical Analysis