次扩散方程的非均匀BDF3-L2方法的误差估计：基于多尺度解分解
Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
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- Yunnan Normal University（云南师范大学）
- Yunnan University of Finance and Economics（云南财经大学）
- Wuhan University（武汉大学）
- Sun Yat-sen University（中山大学）
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中文总结 AI 辅助
针对次扩散方程非均匀L2方法收敛速率与理论预测不符的问题，提出非均匀BDF3-L2方法，结合多尺度解分解与谱截断技术，得到时间收敛阶为$2+\alpha$的误差估计并通过数值实验验证。
中文摘要 AI 辅助
Quan和Wu在《SIAM J Numer Anal》61卷（2023）2106-2132页报道的数值实验表明，次扩散模型的非均匀L2方法观测到的时间收敛速率与理论预测的阶数$3-\alpha$不一致。这种差异说明需要更精细的分析，并推动了次扩散方程非均匀BDF3-L2方法的发展。为处理初始解奇异性，采用多尺度解分解方法分解原解，近似满足带更光滑源项的次扩散模型的更光滑未知变量。然而，所得公式对源项和初始数据包含限制性高阶边界条件。为克服该困难，引入谱截断技术，仅要求数据的正则性稍强，且截断误差可控。建立截断问题解的高阶正则性估计，开发其数值近似的非均匀BDF3-L2方法，据此推导时间收敛阶为$2+\alpha$的严格误差估计。开展数值实验验证理论结果。
英文摘要
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.
