Coefficient Recombination and Error Estimates for the L2 Method for Subdiffusion on Nonuniform Meshes
Abstract
The L2 discretization of the Caputo derivative is attractive for its high-order accuracy, but its analysis on nonuniform temporal meshes is complicated due to the lack of a monotone structure in the discrete coefficients. This difficulty is particularly significant for deriving sharp L2 error estimates in the presence of the typical weak singularity of subdiffusion solutions near the initial time. In this work, we develop a coefficient-recombination framework for the L2 method on nonuniform meshes. Under the adjacent step-ratio condition 0.86≤ ρk≤ 1.16, k≥2, we prove that the recombined L2 coefficients are positive and monotone. A key ingredient is a cross-row recombination technique: with the recombination parameter associated with a fixed time level, all coefficient rows retain positivity and monotonicity, which allows us to close an induction for the L2 error analysis. For solutions exhibiting the standard initial weak singularity, we derive an L2 error estimate on nonuniform meshes satisfying the step-ratio condition. On a modified graded mesh with grading parameter r>1, the estimate yields the convergence rate 𝒪 (N−min{rα, 3−α}), and hence recovers the optimal order 𝒪(N−(3−α)) whenever r≥(3−α)/α. Numerical experiments on random nonmonotone meshes and modified graded meshes confirm the theoretical estimates.
Related
- Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
- Smallest Singular Value Estimates for Nonuniform Fourier Matrices via Periodic Nonuniform Sampling
- Error analysis of a backward L2 method on graded meshes for linear and semilinear subdiffusion equations
- Reverse Heinz type inequality, Mather beta function and coefficient estimates
- Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces
BibTeX
@misc{quan2026coefficient,
title = {{Coefficient Recombination and Error Estimates for the L2 Method for Subdiffusion on Nonuniform Meshes}},
author = {Chaoyu Quan and Shangyu Wang and Shijie Wang and Jiang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39642},
eprint = {2610.39642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39642},
}
RIS
TY - UNPB
TI - Coefficient Recombination and Error Estimates for the L2 Method for Subdiffusion on Nonuniform Meshes
AU - Chaoyu Quan
AU - Shangyu Wang
AU - Shijie Wang
AU - Jiang Yang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39642
UR - https://arxiv.org/abs/2610.39642
AB - The L2 discretization of the Caputo derivative is attractive for its high-order accuracy, but its analysis on nonuniform temporal meshes is complicated due to the lack of a monotone structure in the discrete coefficients. This difficulty is particularly significant for deriving sharp L^2 error estimates in the presence of the typical weak singularity of subdiffusion solutions near the initial time. In this work, we develop a coefficient-recombination framework for the L2 method on nonuniform meshes. Under the adjacent step-ratio condition 0.86≤ ρ_k≤ 1.16, k≥2, we prove that the recombined L2 coefficients are positive and monotone. A key ingredient is a cross-row recombination technique: with the recombination parameter associated with a fixed time level, all coefficient rows retain positivity and monotonicity, which allows us to close an induction for the L^2 error analysis. For solutions exhibiting the standard initial weak singularity, we derive an L^2 error estimate on nonuniform meshes satisfying the step-ratio condition. On a modified graded mesh with grading parameter r>1, the estimate yields the convergence rate 𝒪 (N^−min{rα, 3−α}), and hence recovers the optimal order 𝒪(N^−(3−α)) whenever r≥(3−α)/α. Numerical experiments on random nonmonotone meshes and modified graded meshes confirm the theoretical estimates.
ER -