An α-robust two-grid finite element method with nonuniform L2-1σ scheme for the semilinear Caputo-Hadamard time-fractional diffusion equations involving initial singularity

Considering the initial singularity, a fully discrete two-grid finite element method (FEM) on nonuniform temporal meshes is constructed for the semilinear time-fractional variable coefficient diffusion equations (TF-VCDEs) with Caputo-Hadamard derivative. The nonuniform L${}_{\log ,2-1_{\sigma }}$ formula and two-grid method are employed to discretize the time and space directions, respectively. Through strict theoretical proof, the α-robust stability and optimal $L^2$-norm and $H^1$-norm error analysis for the fully discrete FEM and the two-grid method are obtained, where the error bound does not blow up as $\alpha \to 1^{-}$. To reduce computational costs, a fast two-grid method is constructed by approximating the kernel function with an effective sum-of-exponentials (SOE) technique. Finally, the accuracy and effectiveness of the two-grid method and its associated fast algorithm are verified through two numerical examples.