The Cahn-Hilliard equation has a wide range of applications in many areas of physics and chemistry. To describe the short-range interaction between the solution and the boundary, scientists have constructed dynamical boundary conditions by introducing boundary energy. In this work, the dynamical boundary condition is located on two opposite edges of a square domain and is connected with bulk by a normal derivative. A convex-splitting numerical approach is proposed to enforce the positivity-preservation and energy dissipation, combined with the finite difference spatial approximation. The $\ell^\infty(0,T;H_h^{-1}) \cap \ell^2(0,T;H_h^1)$ convergence analysis and error estimate is theoretically established, with the first order accuracy in time and second order accuracy in space. The bulk and surface discrete mass conservation of the exact solution is required to reach the mean-zero property of the error function, so that the associated discrete $H_h^{-1}$ norm is well-defined. The mass conservation on the physical boundary is maintained by the classic Fourier projection. In terms of the mass conservation in bulk, we introduce a trigonometric auxiliary function based on the truncation error expansion, so that the bulk mass conservation is achieved, and it has no effect on the boundary. The smoothness of trigonometric function makes the Taylor expansion valid and maintains the convergence order of truncation error as well. As a result, the convergence analysis could be derived with a careful nonlinear error estimate.

翻译：Cahn-Hilliard方程在物理学和化学的众多领域中具有广泛应用。为描述溶液与边界之间的短程相互作用，科学家通过引入边界能量构建了动态边界条件。本研究中，动态边界条件位于正方形区域的两条对边上，并通过法向导数与体相连接。本文提出了一种凸分裂数值方法，结合有限差分空间近似，以强制实现保正性与能量耗散。理论上建立了在$\ell^\infty(0,T;H_h^{-1}) \cap \ell^2(0,T;H_h^1)$范数下的收敛性分析与误差估计，时间为一阶精度，空间为二阶精度。精确解的体相与表面离散质量守恒需满足误差函数的均值零特性，从而确保相关离散$H_h^{-1}$范数的良好定义。物理边界上的质量守恒通过经典傅里叶投影保持。针对体相质量守恒，我们基于截断误差展开引入了三角辅助函数，在实现体相质量守恒的同时不影响边界条件。三角函数的平滑性保证了泰勒展开的有效性，并维持了截断误差的收敛阶数。最终，通过精细的非线性误差估计，可推导出收敛性分析结果。