A fully discrete semi-convex-splitting finite-element scheme with stabilization for a Cahn-Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and 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：本文分析了一种用于Cahn-Hilliard交叉扩散系统的全离散、带稳定化的半凸分裂有限元格式。该系统由描述纤维相体积分数和溶质浓度的抛物型四阶方程组成，用于模拟淋巴管形态的预图案化。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，且保持纤维相体积分数的上下界。利用FreeFEM在二维空间进行的数值实验展示了相分离和图案形成过程。