This paper primarily lies in presenting an \(\alpha \)-robust analysis of finite element method for space-time fractional diffusion equation. To this end, we firstly develop finite element approximation to fractional Laplacian and provide a d-dimensional fast Fourier transform (FFT)-based fast algorithm to derive spatial discretization of space-time fractional diffusion problem. Then we study the \(\alpha \)-robust stability and error analyses of full discrete scheme of space-time fractional diffusion problem using L1 scheme on graded temporal meshes. Different from current many existing works, the established stability and error bounds will not blow-up under energy norm as \(\alpha \rightarrow 1^{-}\), and the present error analysis illustrates that a choice of time mesh graded factor \(\gamma =(2-\alpha )/\alpha \) shall yield an optimal rate of convergence \(\mathcal {O}(N^{-(2-\alpha )})\) in temporal direction, where N is the number of temporal meshes. Eventually, some numerical tests are given to show the efficiency and \(\alpha \)-robust behavior of the proposed scheme.

本文主要在于提出一种时空分数扩散方程有限元法的稳健分析。为此，我们首先开发了分数拉普拉斯的有限元近似，并提供了一种基于d维快速傅里叶变换（FFT）的快速算法来导出时空分数扩散问题的空间离散。然后我们研究了在分级时间网格上使用L 1 格式的时空分数扩散问题的全离散格式的\(\alpha \)鲁棒稳定性和误差分析。与当前许多现有工作不同，所建立的稳定性和误差界限在能量范数下不会像\(\alpha \rightarrow 1^{-}\)那样爆炸，并且当前的误差分析表明时间网格分级因子的选择\(\gamma =(2-\alpha )/\alpha \)应在时间方向上产生最佳收敛速度\(\mathcal {O}(N^{-(2-\alpha )})\) ，其中N是时间网格的数量。最后，给出了一些数值测试来显示所提出方案的效率和鲁棒性。