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Structure-preserving semi-convex-splitting numerical scheme for a Cahn-Hilliard cross-diffusion system in lymphangiogenesis

Ansgar Jüngel, Boyi Wang

TL;DR

The paper develops a thermodynamically consistent Cahn–Hilliard cross-diffusion model for lymphangiogenesis and designs a structure-preserving fully discrete finite-element scheme based on stabilized semi-convex-splitting. It proves the existence of discrete solutions, establishes energy stability up to stabilization, ensures solute-mass conservation, and enforces bounds $0\le\phi\le1$ for the fiber phase, while conducting 2D numerical experiments that demonstrate phase separation and pattern formation. The work advances mathematical understanding of cross-diffusion–Cahn–Hilliard systems and provides a robust numerical framework for simulating lymphatic vessel pre-patterning in collagen implants. Potential impact includes improved numerical tools for studying lymphangiogenesis and related biological patterning phenomena, with implications for tissue engineering and vascular network design.

Abstract

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.

Structure-preserving semi-convex-splitting numerical scheme for a Cahn-Hilliard cross-diffusion system in lymphangiogenesis

TL;DR

The paper develops a thermodynamically consistent Cahn–Hilliard cross-diffusion model for lymphangiogenesis and designs a structure-preserving fully discrete finite-element scheme based on stabilized semi-convex-splitting. It proves the existence of discrete solutions, establishes energy stability up to stabilization, ensures solute-mass conservation, and enforces bounds for the fiber phase, while conducting 2D numerical experiments that demonstrate phase separation and pattern formation. The work advances mathematical understanding of cross-diffusion–Cahn–Hilliard systems and provides a robust numerical framework for simulating lymphatic vessel pre-patterning in collagen implants. Potential impact includes improved numerical tools for studying lymphangiogenesis and related biological patterning phenomena, with implications for tissue engineering and vascular network design.

Abstract

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.
Paper Structure (17 sections, 3 theorems, 74 equations, 8 figures, 2 tables)

This paper contains 17 sections, 3 theorems, 74 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model equations
  3. Stabilized semi-convex-splitting scheme
  4. Energy stability and physical bounds
  5. Main results and state of the art
  6. Thermodynamic derivation of the model
  7. Stabilized semi-convex-splitting scheme
  8. Proof of Theorem \ref{['thm.ex']}
  9. Uniqueness of solutions
  10. Regularized problem and regularized energy inequality
  11. Solution to the regularized problem
  12. Uniform bounds
  13. Limit $\delta\to 0$
  14. Numerical experiments
  15. Convergence, energy inequality, and physical bounds
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\sigma>0$ be a given constant, $(\phi_h^0,c_h^0)\in X_h^2$ with $0<\phi_h^0<1$ in ${{\mathbb T}^d}$, and let the time step size $\tau>0$ be sufficiently small. Let $g:{\mathbb R}\to{\mathbb R}$ be nonnegative and $g(0)=0$. Then, for all $n=1,\ldots,N$, there exists a unique solution $(\phi_h^{n where the discrete energy is given by

Figures (8)

  • Figure 1: Initial data $\phi^0$ (left) and $c^0$ (right) used in Section \ref{['sec.num1']}.
  • Figure 2: Discrete energy (left) and extreme values $\min\phi_h^n$ and $\max\phi_h^n$ (right) for various values of the time step $\tau$.
  • Figure 3: Discrete energy (left) and extreme values $\min\phi_h^n$ and $\max\phi_h^n$ (right) for various values of the regularization parameter $\delta$.
  • Figure 4: Extreme values $\min\phi_h^n$ and $\max\phi_h^n$ (left) and extreme values $\min c_h^n$ and $\max c_h^n$ (right) for various values of $\theta_0$ with $\delta=10^{-3}$.
  • Figure 5: Fiber phase fraction $\phi$ at various times $t = 0.2,\, 0.4,\, 0.5,\,0.7,\,0.9,\,5$ with $\varepsilon=0.15$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1: Existence of a discrete solution
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof