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

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.