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

TL;DR

This paper develops and analyzes a time-spectral method for fractional diffusion problems with low regularity data, achieving optimal error estimates and a sharp convergence rate of 1+2α.

Contribution

The paper introduces a new regularity analysis in Besov spaces and establishes optimal error bounds for a combined spectral and finite element method.

Findings

Sharp temporal convergence rate of 1+2α demonstrated

Optimal error estimates achieved for nonsmooth data

New regularity results in Besov spaces for fractional diffusion solutions

Abstract

This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $\alpha$ ($0 < \alpha < 1$). 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 $1+2\alpha$ is shown theoretically and numerically.