Error Analysis of a Fully Discrete Scheme for The Cahn--Hilliard Cross-Diffusion Model in Lymphangiogenesis
Boyi Wang, Naresh Kumar, Jinyun Yuan
TL;DR
The paper addresses the numerical analysis of a stabilized fully discrete finite element scheme for a Cahn–Hilliard cross-diffusion system modeling lymphangiogenesis, where strong cross-coupling and nonlinear diffusion pose stability and convergence challenges. The authors introduce a backward-Euler stabilized scheme, prove discrete energy stability and existence via Brouwer’s fixed-point argument, and derive rigorous error estimates including a novel $L^{\frac{4}{3}}(0,T;W^{1,6/5}(\Omega))$ bound for the chemical potential to achieve convergence to a weak solution. The convergence analysis shows that, as the mesh size $h$ and time step $\tau$ vanish, the numerical solution converges to a weak solution of the continuous model, with optimal rates in time and space under mild regularity. Numerical experiments in two dimensions validate the theoretical results, demonstrating the scheme’s ability to capture phase-separation dynamics characteristic of the Cahn–Hilliard equation. This work provides a rigorous and robust numerical framework for complex cross-diffusion CH systems relevant to lymphangiogenesis and related multiphase phenomena.
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 $L^{\frac{4}{3}}(0,T; L^{\frac{6}{5}}(Ω))$ norm for the chemical potential. This enables a comprehensive convergence analysis, where we derive error estimates in the $L^{\infty}(H^1(Ω))$ and $L^{\infty}(L^2(Ω))$ norms, and establish convergence of the numerical solution in the $L^{\frac{4}{3}}(0,T; W^{1,\frac{6}{5}}(Ω))$ norm. Furthermore, the convergence analysis relies on a uniform bound of the form $\sum_{k=0}^nτ\|\nabla(\cdot)\|_{L^{\frac{6}{5}}}^{\frac{4}{3}}$ to control the chemical potentials, marking a clear departure from the classical $\sum_{k=0}^nτ\|\nabla(\cdot)\|_{L^{2}}^{2}$ 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.