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Enriched Galerkin Method for Navier-Stokes Equations

Chun Song, Minfu Feng

TL;DR

This work develops an Enriched Galerkin (EG) discretization for the stationary incompressible Navier–Stokes equations by blending a continuous velocity space with elementwise bubbles to achieve local conservation while keeping the cost low. A pressure-robust variant (PR-EG) is constructed via a velocity reconstruction operator that maps to the Brezzi–Douglas–Marini space, enabling a stable, robust coupling between velocity and pressure and allowing the viscous term to be discretized with a symmetric interior-penalty method. The authors establish well-posedness under a small-data assumption and derive a priori error estimates: the velocity error in the mesh-dependent energy norm scales as $O(h)$ with a factor depending on $oldsymbol{u}$, and the pressure error in $L^2$ scales as $O(h)$, with corresponding bounds for the velocity in $L^2$ and higher-order terms. Numerical experiments on manufactured and lid-driven cavity tests confirm the theoretical convergence rates and show that the PR-EG scheme accurately captures essential flow structures across viscosities, demonstrating robustness and practical effectiveness for incompressible flow simulation.

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.

Enriched Galerkin Method for Navier-Stokes Equations

TL;DR

This work develops an Enriched Galerkin (EG) discretization for the stationary incompressible Navier–Stokes equations by blending a continuous velocity space with elementwise bubbles to achieve local conservation while keeping the cost low. A pressure-robust variant (PR-EG) is constructed via a velocity reconstruction operator that maps to the Brezzi–Douglas–Marini space, enabling a stable, robust coupling between velocity and pressure and allowing the viscous term to be discretized with a symmetric interior-penalty method. The authors establish well-posedness under a small-data assumption and derive a priori error estimates: the velocity error in the mesh-dependent energy norm scales as with a factor depending on , and the pressure error in scales as , with corresponding bounds for the velocity in and higher-order terms. Numerical experiments on manufactured and lid-driven cavity tests confirm the theoretical convergence rates and show that the PR-EG scheme accurately captures essential flow structures across viscosities, demonstrating robustness and practical effectiveness for incompressible flow simulation.

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 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 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.

Paper Structure

This paper contains 9 sections, 14 theorems, 97 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Problem and Preliminaries
  3. Enriched Galerkin Methods for Navier-Stokes Equations
  4. Well-Posedness of the PR-EG Method and Error Analysis
  5. Well-Posedness of the PR-EG Method
  6. Error Analysis
  7. Numerical Experiments
  8. Example 1: Smooth Polynomial Manufactured Solution
  9. Example 2: Lid-driven cavity flow

Key Result

Lemma 2.1

We define a constant $C_{NE} > 0$, $C$ is a constant $C > 0$, which is also independent of $h$ and $\mu$. So that, we have the following equivalence: where $0 < c_1 < 1$. Furthermore, we can know that $C_{NE}$ has the following bound:

Figures (2)

  • Figure 1: Exact and EG velocity components for Example 1 (smooth polynomial solution).
  • Figure 2: EG approximation of the lid-driven cavity flow: contour plots of $u_{1,h}$, $u_{2,h}$, and $p_h$ on a mesh with $h_0 = 1/32$.

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 15 more