A fully discrete semi-convex-splitting finite-element scheme with stabilization for a degenerate Cahn-Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and the solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFEM illustrate the phase segregation and pattern formation.

翻译：本文分析了一种带有稳定化的完全离散半凸分裂有限元格式，用于求解退化的Cahn-Hilliard交叉扩散系统。该系统由关于纤维相体积分数和溶质浓度的四阶抛物型方程组成，模拟淋巴管形态的预图案化过程。证明了离散解的存在性，并表明该数值格式在稳定化条件下具有能量稳定性，能保持溶质质量守恒，并维持纤维相体积分数的上下界。利用FreeFEM进行的二维空间数值实验展示了相分离与图案形成过程。