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

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.