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H(curl)-based approximation of the Stokes problem with slip boundary conditions

Wietse M. Boon, Ralf Hiptmair, Wouter Tonnon, Enrico Zampa

TL;DR

The paper addresses enforcing Navier slip boundary conditions in a Stokes formulation that seeks velocity in $\mathbf{H}(\mathrm{curl},\Omega)$, by interpreting slip as a Robin boundary condition tied to the boundary Weingarten map. It develops a curl-based, nonconforming discrete scheme stabilized through a curl-preserving lifting $L_h$ and an elliptic projection $\Pi_h$, proving well-posedness and deriving a priori error estimates; a saddle-point reformulation is used for practical pressure computation. Key contributions include stability and convergence results for the nonconforming discretization and a rigorous treatment of the Robin boundary effect in the discrete setting, along with a thorough numerical validation across five 2D test cases showing optimal rates and physically consistent behavior. The work provides a Maxwell-compatible approach to Stokes flow with slip that can be extended to $\mathbf{H}(\mathrm{div})$-based formulations and Dirichlet enforcement via Nitsche methods, with potential applications to elasticity and traction problems. Overall, it offers a robust, theoretically grounded framework for Navier-slip Stokes discretizations in $\mathbf{H}(\mathrm{curl})$ spaces and demonstrates practical viability through comprehensive 2D simulations.

Abstract

The equations governing incompressible Stokes flow are reformulated such that the velocity is sought in the space H(curl). This relaxed regularity assumption leads to conforming finite element methods using spaces common to discretizations of Maxwell's equations. A drawback of this approach, however, is that it is not immediately clear how to enforce Navier-slip boundary conditions. By recognizing the slip condition as a Robin boundary condition, we show that the continuous system is well-posed, propose finite element methods, and analyze the discrete system by deriving stability and a priori error estimates. Numerical experiments in 2D confirm the derived, optimal convergence rates.

H(curl)-based approximation of the Stokes problem with slip boundary conditions

TL;DR

The paper addresses enforcing Navier slip boundary conditions in a Stokes formulation that seeks velocity in , by interpreting slip as a Robin boundary condition tied to the boundary Weingarten map. It develops a curl-based, nonconforming discrete scheme stabilized through a curl-preserving lifting and an elliptic projection , proving well-posedness and deriving a priori error estimates; a saddle-point reformulation is used for practical pressure computation. Key contributions include stability and convergence results for the nonconforming discretization and a rigorous treatment of the Robin boundary effect in the discrete setting, along with a thorough numerical validation across five 2D test cases showing optimal rates and physically consistent behavior. The work provides a Maxwell-compatible approach to Stokes flow with slip that can be extended to -based formulations and Dirichlet enforcement via Nitsche methods, with potential applications to elasticity and traction problems. Overall, it offers a robust, theoretically grounded framework for Navier-slip Stokes discretizations in spaces and demonstrates practical viability through comprehensive 2D simulations.

Abstract

The equations governing incompressible Stokes flow are reformulated such that the velocity is sought in the space H(curl). This relaxed regularity assumption leads to conforming finite element methods using spaces common to discretizations of Maxwell's equations. A drawback of this approach, however, is that it is not immediately clear how to enforce Navier-slip boundary conditions. By recognizing the slip condition as a Robin boundary condition, we show that the continuous system is well-posed, propose finite element methods, and analyze the discrete system by deriving stability and a priori error estimates. Numerical experiments in 2D confirm the derived, optimal convergence rates.
Paper Structure (17 sections, 25 theorems, 103 equations, 5 figures)

This paper contains 17 sections, 25 theorems, 103 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Stokes problem in rotation form with slip boundary condition
  3. Variational formulation of the continuous problem
  4. Functional Setting
  5. The continuous problem
  6. A priori analysis of the discrete problem
  7. Stability
  8. A priori error estimates
  9. Saddle point formulation and error estimates for the pressure
  10. Numerical experiments
  11. A manufactured solution
  12. Fully-driven flow on an annulus
  13. Lid-driven cavity
  14. Slippery semi-circular cavity
  15. Flow around slippery cylinder
  16. ...and 2 more sections

Key Result

Lemma 3.1

$\boldsymbol{\gamma}_{\parallel}:\mathbf{H}^s(\Omega)\to \mathbf{H}^{s-\frac{1}{2}}_{\parallel}(\Gamma)$, $\frac{1}{2} < s \leq \frac{3}{2}$, and $\boldsymbol{\gamma}_{\parallel}:\mathbf{H}(\operatorname{curl}, \Omega)\to \mathbf{H}^{-\frac{1}{2}}_{\parallel}(\Gamma)$ are bounded.

Figures (5)

  • Figure 1: Convergence analysis of (top) $u$ in the $L^2$ and $\mathbf{H}(\operatorname{curl}, \Omega)$ norms, and (bottom) $p$ in the $L^2$ and $H^1$ norms for the experiment as discussed in \ref{['sec:experiment1']}. The results for lowest-order elements are labeled as '' order 1''.
  • Figure 2: Visualization of the computed solution as described in \ref{['sec:Annulus']}. The '' $\mathbf{H}(\operatorname{curl}, \Omega)$'' method correctly produces a rigid body motion. (Left) The white lines represent the streamlines and the colors indicate the magnitude of the velocity field $\mathbf{u}$. (Right) The lines represents the magnitude of the velocity field $\mathbf{u}_h$ on the line segment $c(\gamma)=[1+3\gamma,0]$. Note that the blue and orange lines overlap.
  • Figure 3: Visualization of the computed solution as described in \ref{['sec:LidDrivenCavity']} using 3rd order polynomials on an unstructured mesh with mesh-width $h=0.05$. The domain is a 1-by-1 square. (Left) The white lines represent the streamlines and the colors indicate the magnitude of the velocity field $\mathbf{u}$. (Right) The line represents the magnitude of the discrete velocity $\mathbf{u}_h$ on the line segment $c(\gamma)=[0.5,\gamma]$.
  • Figure 4: Visualization of the computed solution as described in \ref{['sec:SlipperySemiCircularCavity']}. The domain is a half-circle of radius 1 and with center $(0,0)$. We used 3rd order polynomials on an unstructured, curved mesh with mesh-width $h=0.05$. (Left) The white lines represent the streamlines and the colors indicate the magnitude of the velocity field $\mathbf{u}_h$. (Right) The line represents the $x$-component of the approximation of the velocity field $\mathbf{u}$ on the line $x=0$ and $y\in(-1,0)$.
  • Figure 5: Visualization of the computed solution as described in \ref{['sec:FlowAroundSlipperyCylinder']}. The domain is 8-by-8 square with a circle of radius 1 cut out at its center. We used 3rd order polynomials on an unstructured, curved mesh with mesh-width $h=0.25$. (Left) The white lines represent the streamlines and the colors indicate the magnitude of the velocity field $\mathbf{u}$. (Right) Visualization of the computed solution as described in \ref{['sec:FlowAroundSlipperyCylinder']}. The line represents the $x$-component of the produced velocity field $\mathbf{u}_h$ on the the curve $(x, y) = (-\cos(\gamma), \sin(\gamma))$ for $\gamma\in(0,\pi)$.

Theorems & Definitions (56)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 46 more