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Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis

Aparna Bansal, Nicolás Barnafi, Rodolfo Araya, Dwijendra Narain Pandey

TL;DR

This work develops and analyzes an equal-order stabilized finite element method for the stationary incompressible Navier–Stokes equations with slip boundary conditions, employing Nitsche's method to impose slip on complex boundaries. The authors establish well-posedness and stability of the discrete problem for symmetric, skew-symmetric, and incomplete Nitsche variants, and derive rigorous a priori error bounds with optimal convergence under standard regularity assumptions. They also formulate residual-based a posteriori error estimators, proving reliability and efficiency, and validate the theory with extensive numerical experiments including analytic solutions, the lid-driven cavity, adaptive refinement on non-convex domains, and 3D flow around a cylinder. The results demonstrate the method’s robustness, accuracy, and adaptability, offering a practical framework for slip-boundary simulations on complex geometries and adaptive meshes. Future work targets unsteady Navier–Stokes extensions and parameter-robust preconditioning to improve scalability.

Abstract

In this work, we extend the equal-order stabilized scheme discussed in [Franca et al., Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233] to accommodate slip (i.e., Navier) boundary conditions for the stationary Navier-Stokes equations. Our analysis presents a robust formulation for implementing slip boundary conditions using Nitsche's method on arbitrarily complex boundaries. The well-posedness of the discrete problem is established under mild assumptions together with optimal convergence rates for the approximation error. Furthermore, we establish the efficiency and reliability of residual-based a posteriori error estimators for the stationary discrete problem. Several well-known numerical tests validate our theoretical findings. The proposed method fits naturally within the framework of finite element implementation, offering both accuracy and enhanced flexibility in the selection of finite element pairs.

Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis

TL;DR

This work develops and analyzes an equal-order stabilized finite element method for the stationary incompressible Navier–Stokes equations with slip boundary conditions, employing Nitsche's method to impose slip on complex boundaries. The authors establish well-posedness and stability of the discrete problem for symmetric, skew-symmetric, and incomplete Nitsche variants, and derive rigorous a priori error bounds with optimal convergence under standard regularity assumptions. They also formulate residual-based a posteriori error estimators, proving reliability and efficiency, and validate the theory with extensive numerical experiments including analytic solutions, the lid-driven cavity, adaptive refinement on non-convex domains, and 3D flow around a cylinder. The results demonstrate the method’s robustness, accuracy, and adaptability, offering a practical framework for slip-boundary simulations on complex geometries and adaptive meshes. Future work targets unsteady Navier–Stokes extensions and parameter-robust preconditioning to improve scalability.

Abstract

In this work, we extend the equal-order stabilized scheme discussed in [Franca et al., Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233] to accommodate slip (i.e., Navier) boundary conditions for the stationary Navier-Stokes equations. Our analysis presents a robust formulation for implementing slip boundary conditions using Nitsche's method on arbitrarily complex boundaries. The well-posedness of the discrete problem is established under mild assumptions together with optimal convergence rates for the approximation error. Furthermore, we establish the efficiency and reliability of residual-based a posteriori error estimators for the stationary discrete problem. Several well-known numerical tests validate our theoretical findings. The proposed method fits naturally within the framework of finite element implementation, offering both accuracy and enhanced flexibility in the selection of finite element pairs.
Paper Structure (17 sections, 24 theorems, 156 equations, 5 figures, 7 tables)

This paper contains 17 sections, 24 theorems, 156 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The model problem
  3. Discrete stabilized scheme
  4. Well-posedness and stability analysis
  5. Existence of the discrete solution under weak conditions
  6. Existence and uniqueness of the discrete solution under strict conditions
  7. Stability analysis
  8. a priori analysis
  9. A posteriori error estimation for the stationary Navier Stokes problem
  10. Reliability
  11. Efficiency
  12. Numerical experiments
  13. Analytic solution
  14. Lid driven cavity problem
  15. Convergence in the case of adaptive mesh refinement (non-convex domain)
  16. ...and 2 more sections

Key Result

Lemma 1

MR851383MR1318914 \newlabellemma2.10 There are positive constants $\alpha$ and $\beta$, depending only on $\Omega$, such that for all $\boldsymbol{v}, \boldsymbol{w} \in \mathrm{V}$ and $q \in \Pi$, it holds Moreover, for all $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w} \in \mathrm{V}$, it holds $( \boldsymbol{w} \cdot \nabla \boldsymbol{v} , \boldsymbol{v})=-\frac{1}{2}(\nabla \cdot \boldsym

Figures (5)

  • Figure 1: Refined mesh and streamlines with no-slip (top row) and slip (bottom row) boundary conditions at $Re = 5000.$
  • Figure 2: The refined meshes obtained using the adaptive strategy with $\tilde{\theta} = 0.5$, corresponding to $451$, $1912$, and $11146$ degrees of freedom, respectively.
  • Figure 3: Comparison of errors and error indicators for adaptive refinement (left) and uniform refinement (right) with $\tilde{\theta} = 0.5$.
  • Figure 4: Surface view of the final adapted mesh with $\tilde{\theta} = 0.5$ (976,763 elements), showing no-slip (top) and slip (bottom) conditions on the inner cylinder.
  • Figure 5: Velocity magnitude and Pressure isovalues on the final adapted mesh with no-slip (top row) and slip (bottom row) boundary conditions.

Theorems & Definitions (48)

  • Lemma 1
  • Lemma 1
  • Proof 1
  • Remark 3.1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 4.1
  • Lemma 5
  • ...and 38 more