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

Abstract

This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order α (0 < α < 1). The solution regularity in the Sobolev space is revisited, and new regularity results in the Besov space are established. A time-spectral algorithm is developed which adopts a standard spectral method and a conforming linear finite element method for temporal and spatial discretizations, respectively. Optimal error estimates are derived with nonsmooth data. Particularly, a sharp temporal convergence rate 1 + 2α is shown theoretically and numerically.