标题： 变量步长L1格式在具有斜率选择的时间分数阶MBE模型中的相容$L^2$范数收敛性 显示英文标题

标题： Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection

摘要： 研究了带有斜率选择的时间分数阶分子束外延（MBE）模型的变步长L1格式的收敛性。在收敛-可解性-稳定性（CSS）一致的时间步长约束下，建立了变步长L1格式的新型渐近相容的$L^2$范数误差估计。CSS一致条件意味着为收敛所需的最大步长限制与为可解性和稳定性（在某些范数下）所需的最大步长限制具有相同的量级，当小界面参数$\epsilon\rightarrow 0^+$时。据我们所知，这是首次为非线性次扩散问题建立这样的误差估计。渐近相容的收敛性意味着误差估计与经典MBE模型的后向欧拉格式的误差估计相容，当分数阶$\alpha\rightarrow 1^-$时。正如后向欧拉格式可以保持MBE方程的物理性质一样，变步长L1格式也可以保持时间分数阶MBE模型的相应性质，包括体积守恒、变分能量耗散定律和$L^2$范数有界性。数值实验用于支持我们的理论结果。 显示英文摘要

摘要： 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.