Error bounds for time stepping of fractional diffusion equations with non-smooth initial data

Abstract:

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 nth time level tn, 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 for decreasing time is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated using a model 1D problem.