An α-robust and new two-grid nonuniform L2-1σ FEM for nonlinear time-fractional diffusion equation

This paper constructs and analyzes an α-robust and new two-grid finite element method (FEM) with nonuniform L2-$1_{\sigma }$ formula and its fast algorithms for nonlinear time-fractional diffusion equations. The method incorporates a nonuniform L2-$1_{\sigma }$ formula to achieve temporal second-order accuracy and address the initial solution singularity. By employing a spatial two-grid FEM, computational costs are reduced. Utilizing the cut-off technique and an auxiliary function, the condition on the nonlinear term is lessened to meet the local Lipschitz requirement. We further devise the associated fast algorithms for two-grid nonuniform L2-$1_{\sigma }$ FEM. To prevent roundoff errors, we introduce an innovative fast algorithm to precisely calculate the kernel coefficients. An α-robust analysis of the stability and optimal error estimates in terms of $L^2$-norm and $H^1$-norm for the fully discrete scheme is presented. The derived error bound remains stable as the order of the fractional derivative $\alpha \to 1^{-}$. Furthermore, a new two-grid algorithm and its corresponding fast algorithm are proposed to decrease the computational expenses by eliminating redundancy in discrete convolutional summation. Numerical experiments support our theoretical results, confirming that two-grid FEMs offer greater efficiency in comparison to FEM.