A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form

TL;DR

This paper introduces a pressure-robust, parameter-free enriched Galerkin method tailored for the rotational form of steady incompressible Navier-Stokes equations, improving stability and accuracy in computational fluid dynamics.

Contribution

It develops a novel enriched Galerkin method that is pressure-robust and parameter-free, specifically designed for the rotational form of Navier-Stokes equations, with proven theoretical properties and numerical validation.

Findings

Method achieves pressure robustness and convergence.

Numerical experiments confirm effectiveness.

Unique solution under small-data assumption.

Abstract

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.