  We derive unconditionally stable and convergent variable-step BDF2 scheme for
solving the MBE model with slope selection. The discrete orthogonal convolution
kernels of the variable-step BDF2 method is commonly utilized recently for
solving the phase field models. In this paper, we further prove some new
inequalities, concerning the vector forms, for the kernels especially dealing
with the nonlinear terms in the slope selection model. The convergence rate of
the fully discrete scheme is proved to be two both in time and space in $L^2$
norm under the setting of the variable time steps. Energy dissipation law is
proved rigorously with a modified energy by adding a small term to the discrete
version of the original free energy functional. Two numerical examples
including an adaptive time-stepping strategy are given to verify the
convergence rate and the energy dissipation law.
