Conceitos essenciais
The authors introduce modifications to the Scott-Vogelius finite element method to address convergence issues in pressure spaces while maintaining stability.
Resumo
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.
Estatísticas
max{|sin(θi + θi+1)| | 0 ≤ i ≤ Nz}
(−1)ℓq|Kℓ(z)
∥bk−1,z∥2L2(Kℓ)
Citações
"Mesh refinement strategies are proposed in the literature."
"The modified pressure space inherits inf-sup stability and small divergence of velocity."