Core Concepts

Two one-parameter families of quadratic polynomial enrichments are introduced to enhance the accuracy of the classical Crouzeix-Raviart finite element.

Abstract

The paper presents two one-parameter families of quadratic polynomial enrichments for the classical Crouzeix-Raviart finite element. These enrichments are obtained by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions.
The first enrichment family uses a real parameter α > -1 to define enriched linear functionals Fenr_j,α based on Jacobi weight functions. The authors prove that for α ≠ -6/7, the enriched finite element Cα is a valid finite element. They derive explicit expressions for the basis functions ψ_i and ζ_i associated with Cα and construct a quadratic approximation operator Πenr_2,Cα.
The second enrichment family uses a real parameter β > -1 to define enriched linear functionals Genr_j,β based on the center of gravity of the triangle. The authors prove that for any β > -1, the enriched finite element Eβ is a valid finite element. They derive explicit expressions for the basis functions τ_i and ρ_i associated with Eβ and construct a quadratic approximation operator Πenr_2,Eβ.
Numerical experiments are conducted on various test functions and Delaunay triangulations, demonstrating that the proposed enriched finite elements outperform the standard Crouzeix-Raviart finite element in terms of L1-norm error.

Stats

sin(2πx) cos(2πy)^2
1/(x^2 + y^2 + 8)
e^(-81/16((x-0.5)^2 + (y-0.5)^2))^3
sqrt(64 - 81((x - 0.5)^2 + (y - 0.5)^2))/9 - 0.5
e^(x+y)
1/(x^2 + y^2 + 25)

Quotes

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Key Insights Distilled From

by Federico Nud... at **arxiv.org** 04-25-2024

Deeper Inquiries

The proposed enrichment strategies can be extended to higher-order finite elements by incorporating higher-order polynomial functions as enrichment functions. For example, instead of using quadratic polynomials, one can use cubic or higher-degree polynomials to further enhance the accuracy of the finite element method. This extension would involve defining new enriched linear functionals based on the higher-order polynomials and adjusting the basis functions accordingly. Additionally, the enrichment approach can be applied to other types of nonconforming finite elements by adapting the enrichment functions and linear functionals to suit the specific characteristics of those elements. By carefully selecting the enrichment functions and ensuring the unisolvence of the approximation space, the enrichment strategies can be effectively extended to various types of nonconforming finite elements.

One theoretical limitation of the enrichment approach is the potential increase in computational complexity and memory requirements as higher-order polynomials are introduced. This can lead to challenges in solving large-scale problems efficiently. To address this, techniques such as adaptive mesh refinement and hierarchical basis functions can be employed to manage the increased computational demands effectively. Additionally, the theoretical foundation of the enrichment approach should be rigorously validated to ensure the stability and convergence of the numerical solutions.
In practical terms, implementing the enrichment strategies may require specialized software and computational resources, which could be a limitation for researchers or practitioners with limited access to such tools. To overcome this limitation, open-source libraries and software packages that support enriched finite element methods can be utilized. Moreover, collaboration with experts in numerical analysis and computational mathematics can provide valuable insights and guidance in implementing the enrichment approach effectively.

The enhanced accuracy provided by the enriched Crouzeix-Raviart finite element can benefit a wide range of applications and problem domains, including:
Fluid Dynamics: Simulating complex fluid flow phenomena with irregular geometries and discontinuities can benefit from the improved accuracy of the enriched finite element method. Applications include aerodynamics, hydrodynamics, and turbulence modeling.
Structural Mechanics: Analyzing structures with varying material properties, interfaces, and contact conditions can benefit from the enhanced accuracy of the enriched finite element method. This is particularly useful in structural analysis, fracture mechanics, and composite materials.
Geotechnical Engineering: Modeling soil-structure interaction, underground structures, and geomechanical systems can benefit from the enriched finite element method's ability to capture complex behaviors and discontinuities accurately.
Biomedical Engineering: Simulating biological tissues, medical devices, and biomechanical systems can benefit from the improved accuracy of the enriched finite element method in capturing intricate geometries and material properties.
Overall, any application or problem domain that requires precise numerical solutions to partial differential equations and involves complex geometries or discontinuities can benefit from the enhanced accuracy provided by the enriched Crouzeix-Raviart finite element.

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