Enriched Galerkin Method for Navier-Stokes Equations

TL;DR

This paper introduces an enriched Galerkin finite element method for incompressible Navier-Stokes equations that improves accuracy and robustness, especially at low viscosities, through bubble functions and a pressure-robust scheme.

Contribution

The paper develops a novel enriched Galerkin method with bubble functions and a pressure reconstruction operator, providing optimal error estimates and enhanced robustness for incompressible flow simulations.

Findings

Achieves optimal convergence rates for velocity and pressure.

Demonstrates robustness at low viscosities.

Accurately captures flow structures in numerical experiments.

Abstract

This paper presents an enriched Galerkin (EG) finite element method for the incompressible Navier--Stokes equations. The method augments continuous piecewise linear velocity spaces with elementwise bubble functions, yielding a locally conservative velocity approximation while retaining the efficiency of low-order continuous elements. The viscous term is discretized using a symmetric interior penalty formulation, and the divergence constraint is imposed through a stable pressure space. To enhance the robustness of the velocity approximation with respect to the pressure, a reconstruction operator is introduced in the convective and coupling terms, resulting in a pressure-robust scheme whose accuracy does not deteriorate for small viscosities. Both Picard and Newton linearizations are formulated in a fully discrete manner, and the corresponding linear systems are assembled efficiently at each iteration. Optimal a~priori error estimates are established for the velocity in the mesh-dependent energy norm and for the pressure in the $L^2$ norm. Two representative numerical experiments are presented: a smooth manufactured solution and the lid-driven cavity flow. The numerical results confirm the theoretical convergence rates, demonstrating first-order convergence of the velocity in the energy norm, second-order convergence in the $L^2$ norm, and first-order convergence of the pressure. The proposed EG scheme accurately captures characteristic flow structures, illustrating its effectiveness and robustness for incompressible flow simulation.