In this paper, a novel finite element method for solving a modified Cahn–Hilliard–Hele–Shaw system is proposed. The time discretization is based on the convex splitting of the energy functional in the modified Cahn–Hilliard equation, i.e., the high-order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly, respectively. Designing in this way leads to solving a linear system at each time step, which is much efficient compared to solving a nonlinear system resulting from most existing schemes. The proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable. Numerical results are presented to support our theoretical analysis.

本文提出了一种新颖的有限元方法，用于求解改进的Cahn–Hilliard–Hele–Shaw系统。时间离散化基于修正的Cahn-Hilliard方程中能量函数的凸裂变，即化学势中的高阶非线性项和线性项分别被显式和隐式处理。通过这种方式设计，可以在每个时间步求解线性系统，与求解大多数现有方案产生的非线性系统相比，效率很高。所提出的方案被证明是无条件的能量稳定并且对于相位变量而言是最优收敛的。数值结果表明了我们的理论分析。