An \(\alpha \)-robust analysis of finite element method for space-time fractional diffusion equation

This paper primarily lies in presenting an \(\alpha \) $\alpha$ -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 \(\alpha \) $\alpha$ -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 \(\alpha \rightarrow 1^{-}\) $\alpha \to 1^{-}$ , and the present error analysis illustrates that a choice of time mesh graded factor \(\gamma =(2-\alpha )/\alpha \) $\gamma =(2-\alpha )/\alpha$ shall yield an optimal rate of convergence \(\mathcal {O}(N^{-(2-\alpha )})\) $\mathcal{O}(N^{-(2-\alpha )})$ in temporal direction, where N is the number of temporal meshes. Eventually, some numerical tests are given to show the efficiency and \(\alpha \) $\alpha$ -robust behavior of the proposed scheme.