The optimal error analysis of nonuniform L1 method for the variable-exponent subdiffusion model
Wenlin Qiu, Kexin Li, Yiqun Li, Hao Zhang
TL;DR
This paper develops an optimal error analysis for a variable-exponent subdiffusion model by reformulating the problem via a perturbation-based Transform 1 and applying a nonuniform-temporal L1 discretization together with interpolation quadrature for the convolution term. It proves a temporal convergence rate of $O\left(N^{-\min\{2-\alpha_0,\, r\alpha_0\}}\right)$ under nonuniform time steps, with an enhanced rate $O\left(N^{-(2-\alpha_0)}\right)$ when $r\ge\frac{2-\alpha_0}{\alpha_0}$. The analysis covers stability of the time-semidiscrete scheme and extends to fully discrete Galerkin finite elements, yielding a combined temporal-spatial error of $O\left(N^{-\min\{2-\alpha_0,\, r\alpha_0\}} + h^2\right)$. Numerical experiments corroborate the theoretical results, confirming both temporal and spatial convergence rates. The work advances reliable, efficient numerical treatment of heterogeneous diffusion with variable fractional order and nonuniform time stepping, with potential for high-order extensions in future work.
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.