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

TL;DR

This paper introduces an unconditionally stable Gauge-Uzawa finite element scheme for the chemo-repulsion-Navier-Stokes system, combining stability, efficiency, and eliminating the need for initial pressure or artificial boundary conditions.

Contribution

It develops a fully discrete projection framework that is energy stable, avoids artificial pressure conditions, and provides optimal error estimates for key variables.

Findings

The scheme is unconditionally energy stable.

It requires no initial pressure value.

Numerical experiments confirm accuracy and efficiency.

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.