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

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 the canonical and Uzawa--type formulations, while preserving variational consistency. The proposed GU-FEM possesses two notable advantages: (1) it requires no initial pressure value; (2) it avoids artificial pressure boundary conditions, thereby reducing computational overhead. Furthermore, the scheme is shown to be unconditionally energy stable, and optimal error estimates are derived for the cell density, chemical concentration, and fluid velocity. Finally, several numerical experiments are presented to demonstrate the accuracy, stability, and efficiency of the proposed consistent projection finite element method.