In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and ${\mathcal{H}}_{\alpha }^s$ ($s=1,2,3$) spaces independent of time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes. The uniform ${\mathcal{H}}_{\alpha }^3$ bound is obtained on basis of the uniform $H^1$ estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.

本文研究具有多项式非线性的Cahn-Hilliard方程隐式Euler格式的长时间稳定性。中的统一估算$H^{-\mathrm{1个}}$ 和 ${\mathcal{H}}_{\alpha }^s$ （$s=\mathrm{1个}，2，3$对于这种具有可变时间步长的经典方案所产生的数值解，导出了与时间离散步长无关的空间。制服${\mathcal{H}}_{\alpha }^3$ 边界是根据制服获得的 $H^{\mathrm{1个}}$借助均匀的离散Gronwall引理导出离散化学势的估计值。与连续时间动力系统估计值的比较表明，经典的隐式欧拉方法可以完全保留底层系统的长时间行为。借助傅立叶拟谱空间逼近的数值实验也证明了这种长时间的行为。