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.

翻译：我们以斜坡选择方式获得解决MBE模型的无条件稳定且趋同的可变BDF2计划。变量步骤BDF2方法的离散或交替内核最近通常用于解决阶段字段模型。在本文件中,我们进一步证明,在矢量形式方面,内核特别是处理斜坡选择模式中非线性术语的矢量形式方面,出现了一些新的不平等。完全离散方案的趋同率在时间和空间方面都是在可变时间步骤设定下以$L ⁇ 2$为标准的两个时间和空间。能源消散法通过在原自由能源功能的离散版本中增加一个小术语,证明能源消散法与经修改的能量是严格的。提供了两个数字例子,包括适应性时间步战略,以核实汇合率和能量消散法。