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Pressure-robust optimally convergent H(div) finite element method without the commuting diagram property for the steady Oseen equations

Jin Zhang, Xiaowei Liu

TL;DR

This work tackles discretizations of the steady Oseen equations using $H( ext{div})$-conforming finite elements when the commuting diagram property may fail. It introduces a stabilization strategy combining upwind convection handling with a vorticity-based term to attain pressure-robust, optimal-order velocity accuracy (up to $O(h^{k+1/2})$ in suitable norms) even in convection-dominated regimes, leveraging exact de Rham complexes. The analysis applies to element pairs that violate the commuting diagram property, notably Stenberg-type elements, and demonstrates stability, error estimates, and practical efficiency (fewer degrees of freedom) on general meshes. The results broaden the class of viable, efficient divergence-conforming discretizations for high-Reynolds-number incompressible flows.

Abstract

This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The proposed method incorporates vorticity stabilization to ensure optimal-order convergence of the velocity error, especially for convection-dominated cases. As a crucial component of the analysis, exact de Rham and finite element complexes provide a framework whose utility includes establishing velocity error estimates independent of the discrete inf-sup constant. As a representative example, Stenberg finite elements demonstrate the framework's validity and offer several computational advantages: pressure robustness, fewer degrees of freedom than classical RT or BDM elements due to vertex continuity, and convergence without requiring the commuting diagram property. Moreover, the proposed methodology is applicable to a class of finite element pairs that violate the commuting diagram property, thereby offering new possibilities for efficient discretizations of incompressible fluid problems, particularly in high Reynolds number regimes.

Pressure-robust optimally convergent H(div) finite element method without the commuting diagram property for the steady Oseen equations

TL;DR

This work tackles discretizations of the steady Oseen equations using -conforming finite elements when the commuting diagram property may fail. It introduces a stabilization strategy combining upwind convection handling with a vorticity-based term to attain pressure-robust, optimal-order velocity accuracy (up to in suitable norms) even in convection-dominated regimes, leveraging exact de Rham complexes. The analysis applies to element pairs that violate the commuting diagram property, notably Stenberg-type elements, and demonstrates stability, error estimates, and practical efficiency (fewer degrees of freedom) on general meshes. The results broaden the class of viable, efficient divergence-conforming discretizations for high-Reynolds-number incompressible flows.

Abstract

This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The proposed method incorporates vorticity stabilization to ensure optimal-order convergence of the velocity error, especially for convection-dominated cases. As a crucial component of the analysis, exact de Rham and finite element complexes provide a framework whose utility includes establishing velocity error estimates independent of the discrete inf-sup constant. As a representative example, Stenberg finite elements demonstrate the framework's validity and offer several computational advantages: pressure robustness, fewer degrees of freedom than classical RT or BDM elements due to vertex continuity, and convergence without requiring the commuting diagram property. Moreover, the proposed methodology is applicable to a class of finite element pairs that violate the commuting diagram property, thereby offering new possibilities for efficient discretizations of incompressible fluid problems, particularly in high Reynolds number regimes.

Paper Structure

This paper contains 12 sections, 8 theorems, 119 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Oseen equations and finite element methods
  3. Finite element spaces and de Rham complexes
  4. Examples of finite element paris satisfying Assumption \ref{['assumption:Vh+Qh']}
  5. Stabilized $H(\mathrm{div})$-conforming finite element method
  6. Wellposedness of the finite element method and convergence
  7. An error estimate for the velocity
  8. An error estimate for the pressure
  9. Numerical tests
  10. Proof of Lemma \ref{['lem:stability-stenberg']}
  11. (a) Uniform inf-sup stability for $V_h(M) / Q_h(M)$
  12. (b) Inf-sup stability for $V_h / \bar{Q}_h$

Key Result

Theorem 1

Let $\Omega\subset \mathbb{R}^d$ ($d\in \{ 2,3 \}$) be a bounded, Lipschitz domain. For any $\boldsymbol{u}\in H^r(\Omega)^d$ with $r\ge 1$ satisfying $\mathrm{div}\boldsymbol{u}=0$, there exists a potential $\boldsymbol{z}$ (a scalar when $d=2$, or a vector when $d=3$) with components in $H^{r+1}(\ Moreover, the following stability estimate holds where C > 0 is independent of $\boldsymbol{u}$.

Figures (5)

  • Figure 1: Initial mesh level 1
  • Figure 2: $\vvvert \boldsymbol{u}-\boldsymbol{u}_h \vvvert$ and stabilization coefficient $\delta_0$ when $\nu=10^{-6}$
  • Figure 3: vorticity stabilization.
  • Figure 4: upwind and vorticity stabilization.
  • Figure 5: True solution

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 19 more