In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

在本文中，我们介绍并分析了三组分大分子微球复合（MMC）水凝胶模型的有限元数值方案，该模型采用具有Flory-Huggins-deGennes势能的三元Cahn-Hilliard型方程的形式。数值方法是基于多相空间中能量函数的凸凹分解，其中对数和非线性表面扩散项被隐式处理，而凹扩展线性项被显式更新。应用质量集总有限元空间近似，以确保相位变量的正性。反过来，从理论上讲，对于所提出的完全离散的数值方案，保持正性是合理的。此外，还建立了无条件的能量稳定性，这来自于凸度分析。进行了一些数值模拟，以验证所提出方案的准确性和保正性。