A priori and a posteriori error estimates of a space–time Petrov–Galerkin spectral method for time-fractional diffusion equation

Abstract

Time-fractional diffusion equation is an important transport dynamical model for simulating fractal time random walk. This article is devoted to investigating the a priori and a posteriori error estimates for this model equation. A Petrov–Galerkin spectral method is revisited in this paper to address our problem, which the generalized Jacobi functions and Fourier-like basis functions are utilized as basis for constructing efficient and accurate space–time spectral approximations. Rigorous proofs are given for the stability of our spectral scheme. And then the convergence of the proposed method is proved by establishing an a priori error estimate. Specifically, an efficient and reliable a posteriori error estimator is introduced, and we derive that the residual-based error indicator provides an upper bound and a lower bound for the numerical error. Finally, several numerical experiments are provided to examine our theoretical claims.