  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.

Numerical Analysis;Numerical Analysis;Analysis of PDEs