  The positive definiteness of discrete time-fractional derivatives is
fundamental to the numerical stability (in the energy sense) for
time-fractional phase-field models. A novel technique is proposed to estimate
the minimum eigenvalue of discrete convolution kernels generated by the
nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the
fractional Caputo's derivative. The main discrete tools are the discrete
orthogonal convolution kernels and discrete complementary convolution kernels.
Certain variational energy dissipation laws at discrete levels of the
variable-step L1-type methods are then established for time-fractional
Cahn-Hilliard model.They are shown to be asymptotically compatible, in the
fractional order limit $\alpha\rightarrow1$, with the associated energy
dissipation law for the classical Cahn-Hilliard equation. Numerical examples
together with an adaptive time-stepping procedure are provided to demonstrate
the effectiveness of the proposed methods.
