Error analysis of a fully discrete scheme for the Cahn--Hilliard cross-diffusion model in lymphangiogenesis

Source abstract

This paper introduces a stabilized finite element scheme for the Cahn– Hilliard cross-diffusion model, which is characterized by strongly coupled cross-diffusion matrix, 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 and uniqueness 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 norm for the chemical potential, especially in the case of that the time step size τ > 0 and the space mesh size h > satisfy the relation for some constant C > 0. This enables a comprehensive convergence analysis, where we derive error estimates in the L∞(H1(Ω)) and L∞(L2(Ω)) norms, and establish subsequence convergence of the numerical solution in the (Ω)) norm. Furthermore, the convergence analysis relies on a uniform bound of the form to control the chemical potential, marking a clear departure from the classical 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.