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This paper is devoted to the error analysis of a time-spectral algorithm forfractional diffusion problems of order $\alpha$ ($0 < \alpha < 1$). Thesolution regularity in the Sobolev space is revisited, and new regularityresults in the Besov space are established. A time-spectral algorithm isdeveloped which adopts a standard spectral method and a conforming linearfinite element method for temporal and spatial discretizations, respectively.Optimal error estimates are derived with nonsmooth data. Particularly, a sharptemporal convergence rate $1+2\alpha$ is shown theoretically and numerically.

Optimal error estimation of a time-spectral method for fractional diffusion problems with low regularity data