A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form
In this paper, we develop a novel enriched Galerkin (EG) method for the steady incompressible Navier-Stokes equations in rotational form, which is both pressure-robust and parameter-free. The EG space employed here, originally proposed in [1], differs from traditional EG methods: it enriches the first-order continuous Galerkin (CG) space with piecewise constants along edges in two dimensions or on faces in three dimensions, rather than with elementwise polynomials. Within this framework, the gradient and divergence are modified to incorporate the edge/face enrichment, while the curl remains applied only to the CG component, an inherent feature that makes the space particularly suitable for the rotational form. The proposed EG method achieves pressure robustness through a velocity reconstruction operator. We establish existence, uniqueness under a small-data assumption, and convergence of the method, and confirm its effectiveness by numerical experiments.