  The convergence of variable-step L1 scheme is studied for the time-fractional
molecular beam epitaxy (MBE) model with slope selection.A novel asymptotically
compatible $L^2$ norm error estimate of the variable-step L1 scheme is
established under a convergence-solvability-stability (CSS)-consistent
time-step constraint. The CSS-consistent condition means that the maximum
step-size limit required for convergence is of the same order to that for
solvability and stability (in certain norms) as the small interface parameter
$\epsilon\rightarrow 0^+$. To the best of our knowledge, it is the first time
to establish such error estimate for nonlinear subdiffusion problems. The
asymptotically compatible convergence means that the error estimate is
compatible with that of backward Euler scheme for the classical MBE model as
the fractional order $\alpha\rightarrow 1^-$. Just as the backward Euler scheme
can maintain the physical properties of the MBE equation, the variable-step L1
scheme can also preserve the corresponding properties of the time-fractional
MBE model, including the volume conservation, variational energy dissipation
law and $L^2$ norm boundedness. Numerical experiments are presented to support
our theoretical results.
