Pressure-improved Scott-Vogelius type elements

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.