This paper constructs and analyzes an unconditionally optimal two-grid finite element algorithm with \(\alpha \) -robustness on nonuniform time meshes for solving the nonlinear time-fractional Schrödinger equation with variable coefficient and a weak initial singularity. To address the weak initial singularity of the solution, we employ a unified formula that integrates the L1 and L2- \(1_\sigma \) schemes on nonuniform time meshes. A two-grid Galerkin finite element method (FEM) in spatial dimensions is utilized to minimize computational expenses. The current two-grid method avoids imposing time-step restrictions that depend on the spatial-step size and alleviates the limitations on f(u) to a local Lipschitz continuity. The key is to achieve the unconditionally maximum boundedness of the fully discrete numerical solution and its gradient through a novel analysis. This is accomplished by applying the inverse inequality under the condition \(N^{-\mu } \le h^{r+1}\) , while employing the discrete Sobolev embedding inequality when \(N^{-\mu } \ge h^{r+1}\) , in conjunction with the cutoff technique and the auxiliary variable strategy. With the boundedness of the fully discrete solution and its gradient in the maximum norm established, we further demonstrate the unique solvability and unconditional stability of the fully discrete two-grid FEM scheme, deriving its unconditionally optimal error estimate with \(\alpha \) -robustness under the \(H^1\) -norm, free from any temporal-spatial grid-ratio restrictions. The numerical outcomes support the theoretical conclusions and demonstrate that the proposed unconditionally convergent two-grid FEM algorithm surpasses conventional FEM in terms of efficiency.
