Discrete Poincaré and Trace Inequalities for Piece-wise Polynomial Hybridizable Finite Element Spaces

Core Concepts

The author establishes discrete versions of the Poincaré and trace inequalities for hybridizable finite element spaces, which are crucial for the mathematical analysis of partial differential equations solved by modern numerical methods like the hybridizable discontinuous Galerkin (HDG) and hybrid high-order (HHO) methods.

Abstract

The paper focuses on establishing discrete Poincaré and trace inequalities for hybridizable finite element spaces, which are crucial analytical tools for the study of various numerical methods.
Key highlights:

The author introduces analogues of the classical Poincaré and trace inequalities for any pair of piecewise polynomial functions (uh, ûh) in the hybridizable space Xh^k.
The proof relies on bridging the hybridizable spaces with traditional nonconforming spaces using a Crouzeix-Raviart lifting, allowing the use of existing theories.
The derived inequalities enable the author to demonstrate the stability of second-order elliptic equations solved by the hybridizable discontinuous Galerkin (HDG) method under minimal regularity assumptions.
Variants of the Poincaré and trace inequalities are provided specifically for the HDG method, where the energy term involves the vector-valued variable ph rather than the gradient of the scalar variable uh.
The summary provides a comprehensive overview of the key results and their significance in the analysis of hybridizable finite element methods.

Stats

∥uh∥_L2(Ω;Th) ≲ (hK)^2 |uh|_H1(Ω;Th) + hK ∥uh - ûh∥_L2(Ω;∂Th) + ∥LCR_h(ûbar)∥_H1(Ω;Th) + ∫_Ω ûh dx^2
∥uh∥_L2(Ω;Th) ≲ (1 + (hK)^2) ∥ph∥_L2(Ω;Th) + hK ∥uh - ûh∥_L2(Ω;∂Th) + ∫_Ω uh dx^2
∥uh∥_L2(Ω;∂Th) ≲ (1 + hK) ∥ph∥_L2(Ω;Th) + ∥uh - ûh∥_L2(Ω;∂Th) + ∫_Γ ûh ds^2

Quotes

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Key Insights Distilled From

Discrete Poincaré inequality and Discrete Trace inequality in Piece-wise Polynomial Hybridizable Spaces

by Yukun Yue at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19004.pdf

Deeper Inquiries

How can the discrete Poincaré and trace inequalities be extended to more general mesh configurations, such as those with hanging nodes

To extend the discrete Poincaré and trace inequalities to more general mesh configurations, such as those with hanging nodes, we need to consider the impact of these nodes on the continuity and smoothness of the functions within the elements. Hanging nodes introduce discontinuities in the functions across the mesh, which can affect the validity of the inequalities.
One approach to address this challenge is to modify the existing inequalities to account for the presence of hanging nodes. This may involve adjusting the terms related to jumps across faces or boundaries to accommodate the discontinuities introduced by the hanging nodes. Additionally, the shape regularity assumption may need to be reevaluated to ensure that it holds even in the presence of hanging nodes.
By incorporating the specific characteristics of meshes with hanging nodes into the formulation of the inequalities, we can adapt them to apply more broadly to a wider range of mesh configurations, ensuring their relevance and effectiveness in various finite element methods.

What are the implications of these inequalities for the convergence and error analysis of other hybridizable finite element methods, beyond the HDG method

The implications of the discrete Poincaré and trace inequalities developed in this work extend beyond the stability analysis of the HDG method. These inequalities play a crucial role in the convergence and error analysis of other hybridizable finite element methods by providing essential tools for estimating the behavior of the numerical solutions.
In the context of convergence analysis, the Poincaré and trace inequalities can be used to establish the stability and convergence properties of the numerical schemes based on hybridizable spaces. These inequalities help in quantifying the error between the numerical solution and the exact solution, providing insights into the convergence rates and accuracy of the method.
Furthermore, the inequalities can be utilized to derive a priori error estimates for various hybridizable finite element methods, allowing for the assessment of the method's performance in approximating the solution to the underlying partial differential equations. By leveraging these inequalities, researchers can analyze the convergence behavior and optimize the numerical schemes for improved efficiency and accuracy.

Can the techniques developed in this work be adapted to establish similar discrete functional inequalities for other types of nonconforming finite element spaces

The techniques developed in this work for establishing discrete Poincaré and trace inequalities in hybridizable finite element spaces can be adapted to derive similar functional inequalities for other types of nonconforming finite element spaces. By leveraging the principles and methodologies employed in this study, researchers can extend the analysis to different nonconforming spaces and explore their stability and convergence properties.
To establish similar discrete functional inequalities for other nonconforming finite element spaces, researchers can follow a similar approach of bridging the nonconforming spaces with conforming spaces using appropriate lifting operators. By incorporating the specific characteristics and constraints of the nonconforming spaces into the formulation of the inequalities, it is possible to derive stability and error estimates that are tailored to the unique properties of these spaces.
Overall, the techniques and analytical tools developed in this work can serve as a foundation for exploring and establishing discrete functional inequalities in a broader range of nonconforming finite element spaces, enabling a comprehensive analysis of numerical methods across different discretization schemes.