Error Analysis of a Fully Discrete Scheme for The Cahn--Hilliard Cross-Diffusion Model in Lymphangiogenesis

This paper introduces a stabilized finite element scheme for the Cahn--Hilliard cross-diffusion model, which is characterized by strongly coupled mobilities, nonlinear diffusion, and complex cross-diffusion terms. These features pose significant analytical and computational challenges, particularly due to the destabilizing effects of cross-diffusion and the absence of standard structural properties. To address these issues, we establish discrete energy stability and prove the existence of a finite element solution for the proposed scheme. A key contribution of this work is the derivation of rigorous error estimates, utilizing the novel $L^{\frac{4}{3}}(0,T; L^{\frac{6}{5}}(\Omega))$ norm for the chemical potential. This enables a comprehensive convergence analysis, where we derive error estimates in the $L^{\infty}(H^1(\Omega))$ and $L^{\infty}(L^2(\Omega))$ norms, and establish convergence of the numerical solution in the $L^{\frac{4}{3}}(0,T; W^{1,\frac{6}{5}}(\Omega))$ norm. Furthermore, the convergence analysis relies on a uniform bound of the form $\sum_{k=0}^n\tau\|\nabla(\cdot)\|_{L^{\frac{6}{5}}}^{\frac{4}{3}}$ to control the chemical potentials, marking a clear departure from the classical $\sum_{k=0}^n\tau\|\nabla(\cdot)\|_{L^{2}}^{2}$ estimate commonly used in Cahn--Hilliard-type models. Our approach builds upon and extends existing frameworks, effectively addressing challenges posed by cross-diffusion effects and the lack of uniform estimates. Numerical experiments validate the theoretical results and demonstrate the scheme's ability to capture phase separation dynamics consistent with the Cahn--Hilliard equation.