Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche

TL;DR

This work addresses enforcing slip boundary conditions for the Stokes problem by marrying Nitsche's method with a stabilized equal-order finite element discretization. By introducing a stabilization term and allowing a flexible Nitsche formulation via a parameter θ ∈ {−1,0,1}, the authors achieve a discretization that is consistent, has no extra unknowns, and remains stable with an inf-sup constant independent of the viscosity ν. They prove well-posedness and derive optimal a priori error estimates in the natural velocity–pressure norm, with h^k convergence under standard regularity assumptions. Numerical experiments in 2D and 3D, implemented in FEniCS, validate the method's accuracy and robustness for cavity, NACA0012, and cylinder configurations, highlighting its practical viability and adaptability to various finite element pairs.

Abstract

We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.

Figures (10)

• Figure 1: The computed velocity field for Example 1.
• Figure 2: Computational mesh used for the Naca problem.
• Figure 3: Isovalues of the pressure (left) and velocity magnitude (right).
• Figure 4: Zoom, close to the Naca wing, of the velocity field.
• Figure 5: Cylinder problem. Domain and boundary conditions.

Theorems & Definitions (22)

• Theorem 3.1
• proof
• Remark
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4: Consistency