Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the $n$th time level $t_n$, but the error bound includes a factor $t_n^{-1}$ if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as $t_n$ decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.