  In this work, we revisit the adaptive L1 time-stepping scheme for solving the
time-fractional Allen-Cahn equation in the Caputo's form. The L1 implicit
scheme is shown to preserve a variational energy dissipation law on arbitrary
nonuniform time meshes by using the recent discrete analysis tools, i.e., the
discrete orthogonal convolution kernels and discrete complementary convolution
kernels. Then the discrete embedding techniques and the fractional Gr\"onwall
inequality were applied to establish an $L^2$ norm error estimate on nonuniform
time meshes. An adaptive time-stepping strategy according to the dynamical
feature of the system is presented to capture the multi-scale behaviors and to
improve the computational performance.
