Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
Abstract
Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order 3−α. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order 2+α. Numerical experiments are carried out to substantiate the theoretical findings.
Related
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- An implicit multiscale solver for steady kinetic equations based on wave-particle decomposition
- Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise
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BibTeX
@misc{qiu2026error,
title = {{Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition}},
author = {Wenlin Qiu and Kexin Li and Yiqun Li and Hao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02178},
eprint = {2609.02178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02178},
}
RIS
TY - UNPB
TI - Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
AU - Wenlin Qiu
AU - Kexin Li
AU - Yiqun Li
AU - Hao Zhang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02178
UR - https://arxiv.org/abs/2609.02178
AB - Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order 3−α. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order 2+α. Numerical experiments are carried out to substantiate the theoretical findings.
ER -