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

Shuai Su, Xiurong Yan, Qian Zhang

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.

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

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.

Paper Structure

This paper contains 8 sections, 16 theorems, 98 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A pressure-robust and parameter-free enriched Galerkin method
  3. Theoretical analysis
  4. Well-posedness
  5. Error equations
  6. Error estimates
  7. Numerical experiments
  8. Conclusion

Key Result

Lemma 3.1

For any $\boldsymbol{v}=\left\{\boldsymbol{v}_0, v_b\right\} \in \bm V_h^0$, the following inequality holds Therefore, $|\!|\!| \cdot |\!|\!|$ defines a norm in $\bm V_h^0$.

Figures (8)

  • Figure 4.1: Example \ref{['No flow']}: Plots of the numerical velocity.
  • Figure 4.2: Example \ref{['Lid-driven']}: (a) body force $\bm f_2$; (b) difference in the first velocity component computed with $\bm f_1$ and $\bm f_2$; (c) difference in the second velocity component computed with $\bm f_1$ and $\bm f_2$.
  • Figure 4.3: Example \ref{['Lid-driven']}: Streamlines of lid-driven cavity flow at Re = 5000 (left) with $h = 1/100$, Re = 15000 (middle) with $h = 1/200$, and Re = 22000 (right) with $h = 1/250$.
  • Figure 4.4: Example \ref{['BFS_example']}: Streamlines of backward facing step flows with parabolic inlet at Re = 100, 500, and 1000 from top to bottom.
  • Figure 4.5: Example \ref{['BFS_example']}: Streamlines of backward facing step flows with constant inlet at Re = 100, 500, and 1000 from top to bottom.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 2.1: modified gradient su2025novel
  • Definition 2.2: modified divergence su2025novel
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 28 more