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.

翻译：我们为Cahn-Hilliard方程式提出了一个数字近似法,该方程式包含连续的数据同化,以便实现长期的准确性。该方程式使用Cahn-Hilliard方程式第四顺序四级Cahn-Hilliard方程式的内部空间分解法,加上后向的Euler时间分解法。我们证明,对于任意的不准确初始条件而言,该方法是长期稳定且长时间准确的,前提是在模拟中包含足够的数据测量数据。数字实验显示了该方法在基准测试问题上的有效性。