In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized $\alpha$-method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.

在本文中，我们提出了一种非局部算子方法（NOM），用于利用NOM的高阶连续性解决Cahn-Hilliard（CH）方程。该方法是基于加权残差的方法派生的，并在2D和3D中实现。考虑了周期性边界条件和固体壁边界条件。对于这些边界条件，NOM方案中的最高阶分别为2和3。提出的NOM利用可变支持域来进行自适应优化。广义的$\alpha$-方法用于时间积分，并使用Newton-Raphson方法迭代非线性。针对几种二维和三维基准问题证明了该方法的性能。我们还用6阶偏微分导数实现了CH方程，并研究了高阶系数对相场模式演变的影响。