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

TL;DR

This paper develops a stabilized finite element scheme for the complex Cahn--Hilliard cross-diffusion model in lymphangiogenesis, providing rigorous error analysis and convergence proofs despite the model's nonlinear and coupled features.

Contribution

It introduces a novel error analysis framework using an $L^{4/3}$ norm for chemical potential, addressing challenges from cross-diffusion effects and establishing convergence of the numerical scheme.

Findings

The scheme is energy stable and solutions exist.

Error estimates are rigorously derived in multiple norms.

Numerical experiments confirm theoretical convergence and phase separation modeling.

Abstract

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.