In this paper, we derive a novel recovery type a posteriori error estimation of the Crank-Nicolson finite element method for the Cahn--Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator.

翻译：本文针对Cahn--Hilliard方程，推导了Crank-Nicolson有限元方法的一种新型恢复型后验误差估计。为此，我们同时采用了椭圆重构技术和基于三层时间近似的重构技术，从而得到最优的后验误差估计子。我们提出了一种时空自适应算法，将所推导的后验误差估计子作为误差指示器。通过数值实验验证了理论结果，并与基于残差型后验误差估计子的自适应有限元方法进行了比较。