  A new discrete energy dissipation law of the variable-step fractional BDF2
(second-order backward differentiation formula) scheme is established for
time-fractional Cahn-Hilliard model with the Caputo's fractional derivative of
order $\alpha\in(0,1)$, under a weak step-ratio constraint $0.4753\le
\tau_k/\tau_{k-1}<r^*(\alpha)$, where $\tau_k$ is the $k$-th time-step size and
$r^*(\alpha)\ge4.660$ for $\alpha\in(0,1)$.We propose a novel discrete gradient
structure by a local-nonlocal splitting technique, that is, the fractional BDF2
formula is split into a local part analogue to the two-step backward
differentiation formula of the first derivative and a nonlocal part analogue to
the L1-type formula of the Caputo's derivative. More interestingly, in the
sense of the limit $\alpha\rightarrow1^-$, the discrete energy and the
corresponding energy dissipation law are asymptotically compatible with the
associated discrete energy and the energy dissipation law of the variable-step
BDF2 method for the classical Cahn-Hilliard equation, respectively. Numerical
examples with an adaptive stepping procedure are provided to demonstrate the
accuracy and the effectiveness of our proposed method.
