  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.
