SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 219-248, January 2021.
The Cahn--Hilliard equation is a widely used model that describes among others phase-separation processes of binary mixtures or two-phase flows. In recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical treatment of a recent model which introduces an additional Cahn--Hilliard type equation on the boundary as closure for the Cahn--Hilliard equation in the domain [C. Liu and H. Wu, Arch. Ration. Mech. Anal., 233 (2019), pp. 167--247]. By identifying a mapping between the phase-field parameter and the chemical potential inside of the domain, we are able to postulate an efficient, unconditionally energy stable finite element scheme. Furthermore, we establish the convergence of discrete solutions toward suitable weak solutions of the original model. This serves also as an additional pathway to establish existence of weak solutions. Furthermore, we present simulations underlining the practicality of the proposed scheme and investigate its experimental order of convergence.

中文翻译：

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具有动态边界条件的Cahn-Hilliard方程的一种高效收敛的有限元格式

SIAM数值分析学报，第59卷，第1期，第219-248页，2021年1月。
Cahn-Hilliard方程是一种广泛使用的模型，它描述了二元混合物或两相流的相分离过程。近年来，提出并分析了Cahn-Hilliard方程的不同类型的边界条件。在本出版物中，我们关注的是最近模型的数值处理，该模型在边界上引入了一个附加的Cahn-Hilliard型方程，作为域中[C.Hilliard方程的闭合]。刘和H.吴，拱。配给。机甲。Anal。，233（2019），第167--247页]。通过确定相场参数与域内部化学势之间的映射，我们能够提出一种有效的，无条件的能量稳定有限元方案。此外，我们建立了离散解向原始模型的合适弱解的收敛。这也是建立弱解存在的另一条途径。此外，我们目前的仿真强调了该方案的实用性，并研究了其收敛的实验顺序。