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 tntn, but the error bound includes a factor tn−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 tntn 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.