  We propose a numerical approximation method for the Cahn-Hilliard equations
that incorporates continuous data assimilation in order to achieve long time
accuracy. The method uses a C$^0$ interior penalty spatial discretization of
the fourth order Cahn-Hilliard equations, together with a backward Euler
temporal discretization. We prove the method is long time stable and long time
accurate, for arbitrarily inaccurate initial conditions, provided enough data
measurements are incorporated into the simulation. Numerical experiments
illustrate the effectiveness of the method on a benchmark test problem.
