A New H(div div)-Conforming Finite Element Space for the Biharmonic Equation

핵심 개념

A new H(div div)-conforming finite element space is introduced for the biharmonic equation.

초록

A novel H(div div)-conforming finite element is presented, redistributing degrees of freedom to edges and faces. This method enables efficient numerical solutions for the biharmonic equation. The redistribution process involves transferring vertex and normal plane DoFs to faces and edges. By redistributing these DoFs, a hybridizable mixed method with superconvergence is achieved. The proposed finite element space ensures minimal continuity requirements while offering optimal convergence rates for symmetric tensors. Additionally, new weak Galerkin and C0 discontinuous Galerkin methods are derived for the biharmonic equation in all dimensions. The implementation of this method as a generalization of hybridized HHJ methods from 2D to arbitrary dimensions shows promising results.

통계

A novel H(div div)-conforming finite element space is introduced.
Redistribution of degrees of freedom to edges and faces.
Hybridizable mixed method with superconvergence.
Optimal convergence rates for symmetric tensors.
New weak Galerkin and C0 discontinuous Galerkin methods derived.

인용구

"We provide a brief explanation of the redistribution process by examining DoFs of vertex v0."
"The requirement k ≥ 3 can be relaxed to k ≥ 2 by enriching the shape function space."
"Through a hybridization technique, the implementation of the mixed method developed in this paper can be treated as a generalization of hybridized HHJ methods from 2D to arbitrary dimensions."

핵심 통찰 요약

A New Div-Div-Conforming Symmetric Tensor Finite Element Space with Applications to the Biharmonic Equation

by Long Chen,Xu... 게시일 arxiv.org 03-18-2024

https://arxiv.org/pdf/2305.11356.pdf

더 깊은 질문

How does the proposed finite element space compare to existing methods in terms of computational efficiency

The proposed finite element space offers a novel approach to H(div div)-conforming finite elements by redistributing degrees of freedom to edges and faces, thus avoiding the need for super-smoothness. This redistribution allows for more efficient use of computational resources compared to existing methods that rely on higher smoothness requirements. By eliminating the necessity for high-order continuity constraints on lower-dimensional sub-simplices, the new approach simplifies the implementation process and reduces computational complexity.

What challenges might arise when implementing this new approach in practical applications

Implementing this new approach in practical applications may present several challenges. One challenge could be ensuring the proper handling of edge jumps within each element's patch constraint, as these jumps may not vanish across different elements. Additionally, enforcing continuity conditions on normal planes without imposing super-smoothness could require careful consideration and validation to ensure accuracy and stability in numerical simulations. Furthermore, integrating this method into existing software frameworks or workflows may require adjustments to accommodate its unique DoF structure.

How could this research impact advancements in other fields beyond mathematics

This research has the potential to impact advancements in various fields beyond mathematics. The development of a hybridizable mixed method with superconvergence properties for solving the biharmonic equation can have implications in engineering disciplines such as structural mechanics, fluid dynamics, and electromagnetics. The efficient use of computational resources enabled by this new finite element space could lead to faster and more accurate simulations in real-world applications like material design, optimization studies, and predictive modeling. Moreover, the innovative redistribution technique introduced in this research could inspire similar approaches in other scientific domains seeking improved numerical solutions with reduced computational costs.