A proof of convergence is given for bulk--surface finite element semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The semi-discretisation is studied in the weak formulation as a second order system. Optimal-order uniform-in-time error estimates are shown in the $L^2$ and $H^1$ norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second order system. Numerical experiments illustrate the theoretical results.

翻译：在平坦的域内,对卡恩-希利亚德等式与卡恩-希利亚德型动态边界条件的半分解,为散装表面定点元素半分解提供了趋同的证明;在虚弱的配方中将半分解作为第二顺序系统加以研究;在2美元和1美元的规范中显示最佳顺序统一误差估计数;在一致性和稳定性分析的基础上进行误差估计;根据利用第二顺序系统反对称结构的能源估计,在一个抽象的框架内进行稳定性证明;数字实验说明了理论结果。