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

TL;DR

This paper establishes the optimal error estimates for a nonuniform L1 numerical scheme applied to the variable-exponent subdiffusion model, improving convergence rates and validating results with numerical experiments.

Contribution

It introduces an improved convergence analysis for the nonuniform L1 scheme on variable-exponent subdiffusion models, extending previous results and providing optimal error bounds.

Findings

Proves temporal convergence rate $O(N^{- ext{min}igracevert 2- ext{alpha}(0), r ext{alpha}(0)igracevert})$

Improves existing convergence results for $r extgreater rac{2- ext{alpha}(0)}{ ext{alpha}(0)}$

Numerical experiments confirm theoretical error estimates.

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 $O(N^{-\min\{2-\alpha(0), r\alpha(0)\}})$ under the nonuniform mesh, which improves the existing convergence results in [Zheng, CSIAM T. Appl. Math. 2025] for $r\geq \frac{2-\alpha(0)}{\alpha(0)}$. Numerical results are presented to substantiate the theoretical findings.