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The basic equation for a pressure field in fluid mechanics captures the balance of forces within any segment of fluid, providing a foundational understanding of how pressure changes within fluids under various forces. Generally, two main types of forces act on any part of a fluid: surface forces and body forces. Surface forces arise from pressure differences across points within the fluid, which result in net forces that can vary depending on the local pressure gradient. Body forces, on the...
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Updated: May 7, 2025

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Pressure-improved Scott-Vogelius type elements.

Nis-Erik Bohne1, Benedikt Gräßle1, Stefan A Sauter1

  • 1Institut für Mathematik, Universität Zürich, Winterthurerstr 190, 8057 Zürich, Switzerland.

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|December 31, 2024
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Summary
This summary is machine-generated.

A new modification strategy improves the Scott-Vogelius element for Stokes equations. This method ensures optimal pressure convergence rates while maintaining stability, addressing issues with critical vertices.

Keywords:
Inf-sup stabilityMass conservationScott–Vogelius elementshp finite elements

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Area of Science:

  • Computational fluid dynamics
  • Numerical analysis
  • Finite element methods

Background:

  • The Scott-Vogelius element is widely used for discretizing Stokes equations, offering inf-sup stability and divergence-free velocity approximations.
  • A known limitation is the deterioration of pressure convergence rates near critical vertices in the domain's triangulation.
  • Existing modifications, like the pressure-wired Stokes element, also face challenges with these critical vertices.

Purpose of the Study:

  • To introduce a novel modification strategy for pressure spaces used with the Scott-Vogelius element.
  • To address the convergence rate issues of discrete pressure in the presence of critical vertices.
  • To maintain inf-sup stability while achieving optimal pressure convergence rates.

Main Methods:

  • Development of a simple modification strategy for pressure spaces.
  • Analysis of the modified element's stability properties.
  • Investigation of the convergence rates for discrete pressure approximations.

Main Results:

  • The proposed modification strategy preserves the essential inf-sup stability of the finite element.
  • The strategy effectively resolves the issue of deteriorating pressure convergence rates at critical vertices.
  • Optimal convergence rates for the discrete pressure are achieved.

Conclusions:

  • The introduced modification offers a robust solution for enhancing the performance of the Scott-Vogelius element in Stokes equation discretizations.
  • This approach provides a practical method for improving numerical accuracy in fluid dynamics simulations, particularly in complex geometries.
  • The strategy is applicable to both the standard Scott-Vogelius element and its recent variants.