  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.
