Nonuniform L1/spectral element algorithm for the time fractional diffusion equation

Abstract

In this paper, an efficient numerical scheme is constructed for the time fractional diffusion equation with temporal Caputo derivative of order $\nu \in (0,1)$ and classical Laplacian in space. To deal with the weak regularity of the solution at the initial time, the L1 formula on graded meshes is applied in temporal approximation. For the spatial discretization, the spectral element method is utilized. Unconditional stability and convergence of the proposed scheme are rigorously proved. Implementation details are presented as well. Furthermore, numerical experiment is carried out which supports the theoretical analysis.