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Unconditionally stable Gauge-Uzawa finite element schemes for the chemo-repulsion-Navier-Stokes system

Chenyang Li, Ping Lin, Haibiao Zheng

TL;DR

The authors develop a first-order, fully discrete Gauge–Uzawa finite element method for the chemo-repulsion-Navier-Stokes system, achieving unconditional energy stability and unique solvability without pressure boundary conditioning. By introducing an auxiliary variable for the chemical gradient, they formulate a decoupled scheme that preserves the energy dissipation structure and obtain optimal error estimates for density, concentration, and velocity. Theoretical analysis is complemented by comprehensive numerical experiments that confirm convergence rates, stability, and the expected chemo-repulsion behavior. The framework provides a robust, efficient tool for simulating coupled chemotaxis-fluid interactions with rigorous guarantees.

Abstract

This paper investigates a Gauge-Uzawa finite element method (GU-FEM) for the two-dimensional chemo-repulsion-Navier-Stokes (CRNS) system. The proposed approach establishes a fully discrete projection framework that integrates the advantages of both canonical and Uzawa-type formulations while preserving variational consistency. The method possesses two notable advantages: (1) it requires no initial pressure value; (2) it avoids artificial pressure boundary conditions and thus reduces computational cost. Furthermore, the scheme is shown to be unconditionally energy stable, and we establish unique solvability together with optimal error estimates for cell density, chemical concentration, and fluid velocity. Finally, several numerical experiments are provided to validate the accuracy, stability, and efficiency of the proposed method.

Unconditionally stable Gauge-Uzawa finite element schemes for the chemo-repulsion-Navier-Stokes system

TL;DR

The authors develop a first-order, fully discrete Gauge–Uzawa finite element method for the chemo-repulsion-Navier-Stokes system, achieving unconditional energy stability and unique solvability without pressure boundary conditioning. By introducing an auxiliary variable for the chemical gradient, they formulate a decoupled scheme that preserves the energy dissipation structure and obtain optimal error estimates for density, concentration, and velocity. Theoretical analysis is complemented by comprehensive numerical experiments that confirm convergence rates, stability, and the expected chemo-repulsion behavior. The framework provides a robust, efficient tool for simulating coupled chemotaxis-fluid interactions with rigorous guarantees.

Abstract

This paper investigates a Gauge-Uzawa finite element method (GU-FEM) for the two-dimensional chemo-repulsion-Navier-Stokes (CRNS) system. The proposed approach establishes a fully discrete projection framework that integrates the advantages of both canonical and Uzawa-type formulations while preserving variational consistency. The method possesses two notable advantages: (1) it requires no initial pressure value; (2) it avoids artificial pressure boundary conditions and thus reduces computational cost. Furthermore, the scheme is shown to be unconditionally energy stable, and we establish unique solvability together with optimal error estimates for cell density, chemical concentration, and fluid velocity. Finally, several numerical experiments are provided to validate the accuracy, stability, and efficiency of the proposed method.

Paper Structure

This paper contains 17 sections, 9 theorems, 177 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. An equivalent system
  4. Gauge--Uzawa finite element method
  5. Unconditional stability
  6. Existence and uniqueness of the First-order Gauge-Uzawa algorithm
  7. Convergence analysis
  8. Reduced rate of convergence
  9. The proof for $i=1$
  10. The proof for $i=N$
  11. Optimal error estimates
  12. Numerical experiences
  13. Convergence test
  14. Stability test
  15. The chemo-repulsion behavior
  16. ...and 2 more sections

Key Result

Lemma 2.1

heyinnian2015 Let $C_0, a_n, b_n, d_n$ be nonnegative numbers with integer $n \ge 0$ such that then there holds that

Figures (7)

  • Figure 5.1: Convergence history of $(\mathbf{u},p,\eta,c)$ for different $\tau$.
  • Figure 5.2: Convergence history of $(\mathbf{u},p,\eta,c)$ for different $h$.
  • Figure 5.3: Time evolution of the free energy functional $\mathcal{E}^3$
  • Figure 5.4: Chemical concentration at $k=1,10,15,20,25,30$.
  • Figure 5.5: Cell density at $k=1,10,15,20,25,30$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Div-grad inequality
  • Remark 2.1
  • Remark 2.2
  • proof
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • ...and 9 more