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Development of H(div)-conforming finite element tensors based on geometric decomposition.

Abstract

This work presents a unified construction of H(div)-conforming finite element tensors, including vector div element, symmetric div matrix element, and traceless div matrix element. The approach involves decomposing the tensor at each sub-simplex into tangential and normal components. Boundary degrees of freedom are explored to discover various finite elements that are H(div)-conforming and satisfy the discrete inf-sup condition. An explicit basis for the constraint tensor space is established. The content delves into Hilbert complexes' role in numerical methods for partial differential equations and introduces a systematic approach to derive new complexes from well-understood differential complexes like the de Rham complex. The article discusses the algebraic operator sk,n−1 and its role in deriving new complexes. It also explores the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex.

Stats

n = 2, r = 1 (Lemma 3.9)
r ≥ 2 (Lemma 3.9)
m = -1, m = 0, m = n - 2 (Lemma 3.8)
r ≥ 1 (Proposition 3.10)

Quotes

"The developed finite element spaces are H(div)-conforming and satisfy the discrete inf-sup condition."
"A significant aspect of our contribution is the expansion of geometric decomposition techniques to effectively manage tensors subjected to specific constraints."
"An explicit basis for the constraint tensor space is also established."

Key Insights Distilled From

by Long Chen,Xu... at **arxiv.org** 03-19-2024

Deeper Inquiries

The geometric decomposition approach plays a crucial role in managing tensors with linear constraints by providing a systematic way to decompose the tensor space into tangential and normal components. This decomposition allows for explicit bases for finite elements, making it easier to construct H(div)-conforming finite element spaces. By breaking down the tensor at each sub-simplex into its tangential and normal components, the method ensures that both constraints and normal continuity are satisfied simultaneously. This approach simplifies the construction of finite elements with specific linear constraints, such as symmetry or tracelessness, by providing a clear framework for handling these requirements.

The development of H(div)-conforming finite element spaces has significant implications for practical applications in various fields such as computational mechanics, fluid dynamics, electromagnetics, and structural analysis. These finite element spaces ensure that numerical methods used to solve partial differential equations maintain stability and accuracy while satisfying divergence-free conditions on vector fields. In practical applications like fluid flow simulations or electromagnetic field modeling, maintaining divergence-free properties is essential for capturing physical phenomena accurately.
Additionally, the discrete inf-sup condition established for these H(div)-conforming finite element spaces guarantees stability and robustness of numerical solutions when solving complex problems involving vector fields subject to divergence constraints. The explicit basis of constraint tensor space provides a foundation for developing efficient algorithms that can handle challenging engineering problems effectively.
Overall, the development of H(div)-conforming finite element spaces enhances the reliability and accuracy of numerical simulations in real-world applications where preserving divergence properties is critical.

These findings can be extended to more complex geometries or higher dimensions by adapting the geometric decomposition approach to suit different scenarios. For more complex geometries beyond simplexes, such as irregular domains or curved surfaces, an adaptive refinement strategy can be employed to handle non-standard shapes efficiently.
In higher dimensions, extending this approach involves generalizing the t-n decomposition technique to manage tensors subjected to specific constraints in multidimensional spaces. By incorporating additional dimensions into the decomposition process while ensuring compatibility with existing frameworks like BDM elements or Stenberg elements in 3D cases discussed earlier.
Furthermore, exploring hierarchical geometric decompositions tailored for higher-dimensional structures would enable researchers to address challenges related to tensor management across multiple dimensions effectively.

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