The optimal error analysis of nonuniform L1 method for the variable-exponent subdiffusion model

Abstract: This work investigates the optimal error estimate of the fully discrete scheme for the variable-exponent subdiffusion model under the nonuniform temporal mesh. We apply the perturbation method to reformulate the original model into its equivalent form, and apply the L1 scheme as well as the interpolation quadrature rule to discretize the Caputo derivative term and the convolution term in the reformulated model, respectively. We then prove the temporal convergence rates $O(N^{-\min{2-α(0), rα(0)}})$ under the nonuniform mesh, which improves the existing convergence results in [Zheng, CSIAM T. Appl. Math. 2025] for $r\geq \frac{2-α(0)}{α(0)}$. Numerical results are presented to substantiate the theoretical findings.