Efficient Finite Element Method for Stokes Equations with Slip Boundary Conditions using Stabilized Nitsche Approach

Core Concepts

The authors present a simple and efficient finite element method for solving the Stokes equations with slip boundary conditions, using a stabilized Nitsche approach. The method is consistent, introduces no extra unknowns, and can be easily implemented.

Abstract

The key highlights and insights of the content are:

The authors discuss how slip conditions for the Stokes equation can be handled using the Nitsche method, for a stabilized finite element discretization.

The emphasis is 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 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.

The authors compare their approach with previous methods based on Lagrange multipliers, penalty techniques, and mixed formulations. They highlight the advantages of the Nitsche-based approach in terms of simplicity, consistency, and robustness.

The stability analysis relies on the introduction of proper stabilization terms, allowing to prove stability with an inf-sup constant independent of the fluid viscosity.

Numerical experiments on 2D and 3D benchmark problems demonstrate the simplicity, flexibility, and accuracy of the proposed method.

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Key Insights Distilled From

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

by Rodolfo Aray... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08810.pdf

Deeper Inquiries

How does the performance of the proposed Nitsche-based method compare to other state-of-the-art techniques for solving the Stokes equations with slip boundary conditions, in terms of computational efficiency and accuracy

The proposed Nitsche-based method for solving the Stokes equations with slip boundary conditions offers several advantages compared to other techniques. In terms of computational efficiency, the method demonstrates stability and convergence rates that are independent of the fluid viscosity, as proven in the stability analysis. This independence allows for consistent performance across a wide range of viscosities, enhancing the method's robustness. Additionally, the method introduces proper stabilization terms that contribute to the overall accuracy of the solution. The numerical experiments conducted in the study illustrate the simplicity, flexibility, and accuracy of the method, showcasing its competitive performance in terms of computational efficiency and accuracy.

Can the Nitsche-based approach be extended to handle more complex slip boundary conditions, such as those involving tangential components of the velocity (e.g., Navier slip law)

The Nitsche-based approach presented in the study can indeed be extended to handle more complex slip boundary conditions, including those involving tangential components of the velocity, such as the Navier slip law. By incorporating appropriate modifications to the Nitsche terms and stabilization techniques, the method can be adapted to enforce slip conditions that involve tangential components of the velocity. This extension would involve adjusting the formulation of the Nitsche terms to accurately capture the behavior of the velocity field along the boundary with tangential slip conditions. The flexibility of the Nitsche method allows for such adaptations to handle diverse boundary conditions effectively.

What are the potential applications of the Nitsche-based Stokes solver beyond fluid mechanics, for example in the context of solid mechanics or multiphysics problems involving slip interfaces

The Nitsche-based Stokes solver, beyond its applications in fluid mechanics, holds significant potential for various other fields, including solid mechanics and multiphysics problems involving slip interfaces. In solid mechanics, the method can be utilized to solve problems related to contact mechanics, where slip conditions at interfaces play a crucial role in determining the behavior of the system. By extending the Nitsche approach to solid mechanics problems, researchers can accurately model and analyze contact interactions with slip conditions, leading to more realistic simulations. Furthermore, in multiphysics problems involving slip interfaces, such as fluid-structure interaction or thermal-fluid systems, the Nitsche-based solver can provide a robust framework for handling the coupling between different physical phenomena across slip boundaries. This versatility highlights the broad applicability of the Nitsche method beyond traditional fluid dynamics simulations.