Fast and accurate solvers for time-space fractional diffusion problem with spectral fractional Laplacian

This paper develops fast and accurate linear finite element method and fourth-order compact difference method combined with matrix transfer technique to solve high dimensional time-space fractional diffusion problem with spectral fractional Laplacian in space. In addition, a fast time stepping $L1$ scheme is used for time discretization. We can exactly evaluate fractional power of matrix in the proposed schemes, and perform matrix-vector multiplication by directly using a discrete sine transform and its inverse transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory. Further, we address the convergence analyses of full discrete scheme based on two types of spatial numerical methods. Finally, ample numerical examples are delivered to illustrate our theoretical analyses and the efficiency of the suggested schemes.