We establish a uniform-in-scaling error estimate for the asymptotic preserving scheme proposed in \cite{XW21} for the L\'evy-Fokker-Planck (LFP) equation. The main difficulties stem from not only the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling $\epsilon$: in the regime where $\epsilon$ is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where $\epsilon$ is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.

翻译：我们针对文献\cite{XW21}中提出的Lévy-Fokker-Planck（LFP）方程的渐近保持格式，建立了关于尺度参数的一致误差估计。主要困难不仅来源于尺度参数与数值参数之间的相互作用，还源于平衡态尾部衰减缓慢。为解决这些问题，我们根据尺度参数$\epsilon$的相对大小对参数区域进行划分：在$\epsilon$较大的情形下，设计加权范数以缓解肥尾问题；而在$\epsilon$较小的情形下，证明LFP方程以显式收敛率强收敛至其分数阶扩散极限。该方法将传统AP估计推广至一致界无法获得的场景。我们的结果适用于任意维度及分数阶幂的整个取值范围。