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

TL;DR

This paper analyzes the time-stepping error bounds for a fractional diffusion equation with non-smooth initial data, showing how initial data smoothness affects accuracy and generalizing classical heat equation results.

Contribution

It provides new error bounds for the discontinuous Galerkin method applied to fractional diffusion problems, accounting for initial data smoothness.

Findings

Error bound includes a factor t_n^{-1} for non-smooth initial data

Smoother initial data results in milder error growth over time

Error bounds generalize classical heat equation results

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 \$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.