Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. (English) Zbl 1349.65469
Summary: 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.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 33B30 | Higher logarithm functions |
| 35R11 | Fractional partial differential equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
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