Enhancing Crouzeix-Raviart Finite Element with Quadratic and Cubic Polynomial Enrichments

核心概念

The authors introduce quadratic and cubic polynomial enrichments to enhance the accuracy of the classical Crouzeix-Raviart finite element, presenting necessary conditions for their validity.

要約

In this paper, quadratic and cubic polynomial enrichments are introduced to improve the accuracy of the Crouzeix-Raviart finite element. The enriched elements add weighted line integrals as degrees of freedom, resulting in more accurate approximations. The study provides explicit expressions for basis functions and introduces new approximation operators based on these enrichments. Numerical results show a significant improvement over traditional methods, confirming the effectiveness of the enrichment strategy.

統計

For each case, we respectively add three and seven weighted line integrals as enriched degrees of freedom.
The weight function is defined as t^(α-1/2)(1-t)^(α-1/2).
The normalized coefficient K is defined as 1/(22α).
The polynomial p2 used for defining enriched degrees of freedom covers a wide class of classical orthogonal polynomials.
Varying parameters α and β allows coverage of various classical orthogonal polynomials.

引用

"The numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy."

抽出されたキーインサイト

A new quadratic and cubic polynomial enrichment of the Crouzeix-Raviart finite element

by Francesco De... 場所 arxiv.org 03-12-2024

https://arxiv.org/pdf/2403.05844.pdf

深掘り質問

How do these polynomial enrichments impact computational efficiency compared to traditional methods

The polynomial enrichments proposed in the context significantly impact computational efficiency compared to traditional methods by providing more accurate approximations. By introducing quadratic and cubic polynomial enrichments to the Crouzeix-Raviart finite element, the accuracy of solutions is improved, leading to better convergence rates and reduced errors. This enhancement in accuracy can result in fewer iterations required for convergence, ultimately saving computational time and resources.

What are potential limitations or drawbacks of using higher-order polynomial enrichments in finite element analysis

While higher-order polynomial enrichments offer improved accuracy and convergence properties, there are potential limitations or drawbacks to consider when using them in finite element analysis. One limitation is the increased complexity introduced by higher-order polynomials, which can lead to larger system matrices that require more memory and computational power to solve. Additionally, higher-order enrichments may introduce more degrees of freedom per element, potentially increasing the overall system size and computation time. Moreover, as the order of enrichment increases, numerical instabilities may arise if not properly controlled or accounted for.

How can these findings be applied to other types of finite elements or numerical methods

The findings from these polynomial enrichments can be applied beyond just the Crouzeix-Raviart finite element method. The concept of enhancing traditional finite elements with higher-order polynomials can be extended to other types of finite elements used in numerical methods like Galerkin methods or spectral methods. By incorporating quadratic or cubic polynomial enrichments into different types of finite elements, researchers can achieve similar improvements in accuracy and efficiency across various applications such as fluid dynamics simulations, structural mechanics analyses, electromagnetics modeling, and many others. These enriched elements have the potential to provide more precise solutions while maintaining computational efficiency across a wide range of numerical simulations.