Pressure-Improved Scott-Vogelius Finite Element Modifications

Temel Kavramlar

The authors introduce modifications to the Scott-Vogelius finite element method to address convergence issues in pressure spaces while maintaining stability.

Özet

The paper discusses enhancements to the Scott-Vogelius finite element method for solving Stokes equations. It addresses convergence rate deterioration in discrete pressure spaces due to critical vertices. The proposed modifications aim to maintain inf-sup stability and improve the accuracy of pressure approximation. The study focuses on mesh robustness and optimal convergence rates for discrete solutions.
The authors introduce a simple modification strategy that preserves inf-sup stability while improving the convergence rate of the pressure space. They propose a postprocessing step applied to the original solution, ensuring optimal approximation properties. The paper organizes its content into sections detailing the problem setup, modifications for both Scott-Vogelius and pressure-wired Stokes elements, and theoretical proofs supporting their approach.
Key points include defining critical vertices affecting pressure approximation, introducing modified pressure spaces, proving inf-sup stability inheritance, and demonstrating optimal convergence rates. The study emphasizes simplicity in implementation without increasing computational complexity or altering element dimensions.

İstatistikler

max{|sin(θi + θi+1)| | 0 ≤ i ≤ Nz}
(−1)ℓq|Kℓ(z)
∥bk−1,z∥2L2(Kℓ)

Alıntılar

"Mesh refinement strategies are proposed in the literature."
"The modified pressure space inherits inf-sup stability and small divergence of velocity."

Önemli Bilgiler Şuradan Elde Edildi

Pressure-improved Scott-Vogelius type elements

by Nis-... : arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04499.pdf

Daha Derin Sorular

How do these modifications impact computational efficiency

The modifications introduced in the paper impact computational efficiency in several ways. Firstly, by improving the approximation properties of the pressure space, the overall accuracy of the numerical solution is enhanced. This can lead to faster convergence rates and reduced error in the results obtained from simulations. Additionally, by ensuring that the discrete pressure converges at an optimal rate, it may reduce the need for excessive mesh refinement or post-processing steps, thereby saving computational resources.

What are potential limitations or drawbacks of this approach

One potential limitation or drawback of this approach could be related to implementation complexity. While the modifications aim to improve approximation properties and convergence rates, they may require additional computational overhead for evaluating new functionals or constraints on certain vertices. This could increase algorithmic complexity and potentially slow down computations.
Another limitation could be related to generalizability across different types of problems or geometries. The specific assumptions made in defining functionals fz and choosing Jz might not be universally applicable and may need adjustments based on specific problem characteristics.

How could these findings be applied to other types of finite element methods

These findings on modifying pressure spaces for improved convergence rates can be applied to other types of finite element methods as well. By carefully selecting appropriate functionals or constraints tailored to specific problem domains, similar enhancements in approximation properties can be achieved across various discretization schemes.
For instance, these insights could inform developments in hp-finite elements where higher-order polynomials are used for both velocity and pressure approximations. By incorporating similar modification strategies focused on optimizing convergence rates for pressures within hp-elements, researchers can enhance accuracy while maintaining computational efficiency.
Additionally, these findings could inspire advancements in adaptive mesh refinement techniques by guiding decisions on when and where such modifications are most beneficial based on local singularities or critical points within a mesh structure.