The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the 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, H. Wu, Arch. Ration. Mech. An., 2019]. 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 towards 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.
翻译:Cahn-Hilliard等式是一种广泛使用的模式,除其他外,它描述了二进制混合物或两阶段流的分解过程。近年来,提出了Cahn-Hillard等式的不同类型的边界条件并进行了分析。在本出版物中,我们关注最近一个模型的数值处理,该模型在边界上引入了另外一种Cahn-Hilliard等式,作为Cahn-Hilliard等式在域的封闭[C. Liu, H. Wu, Arch. Ration. Mech. An., 2019]。通过在区域内的分野参数和化学潜力之间绘制地图,我们能够假设一个高效、无条件能源稳定的有限要素计划。此外,我们建立了离散解决办法的趋同,以找到原始模型的合适弱性解决办法。这也成为确定存在薄弱解决办法的又一条途径。此外,我们提出模拟,强调拟议办法的实际性,并调查其实验性趋同顺序。