Finite Element Complexes

有限元复合体

• 批准号: 2309785
• 负责人: Long Chen
• 金额: $ 40.13万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309785&HistoricalAwards=false
• 关键词: Finite; Element; Complexes

This project focuses on enhancing our ability to solve complex mathematical problems that have practical applications in areas such as engineering and physics. The aim is to create a unified framework that will better understand and tackle these problems, using cutting-edge computational techniques. In addition to advancing scientific knowledge, the project has a strong emphasis on education and promoting diversity by training the next generation of computational mathematicians. By developing easy-to-use tools and sharing knowledge through various platforms, the project aims to benefit society at large, improving our understanding of complex systems and enabling the development of innovative solutions in multiple fields. The project will include training of at least two graduate students. The technical aspect of this project employs a powerful approach called Finite Element Exterior Calculus (FEEC), which has already proven effective in analyzing the stability of finite element discretizations and in revealing new finite elements for solving various partial differential equations (PDEs). The goal is to extend the range of FEEC to design finite element spaces for tensors and construct additional finite element complexes. The project will utilize the Bernstein-Gelfand-Gelfand (BGG) construction to systematically develop finite element complexes for various applications, such as Hessian complex, elasticity complex, and divdiv complex. Moreover, this project seeks to unify non-conforming finite element methods, weak Galerkin element methods, and virtual element methods using weak stability and distributional finite element complexes, thus expanding the finite element periodic table. Research topics include smooth finite element spaces for scalar functions, div-conforming face elements for tensors, finite element de Rham and Stokes complexes, discrete BGG construction, weak div and divdiv stability, and distributional finite element complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目着重于提高我们解决在工程和物理等领域具有实际应用的复杂数学问题的能力。目的是创建一个统一的框架，该框架将使用前沿计算技术更好地理解和解决这些问题。除了促进科学知识之外，该项目还通过培训下一代计算数学家培训教育和促进多样性。通过开发易于使用的工具并通过各种平台共享知识，该项目旨在使社会受益，从而提高我们对复杂系统的理解，并能够在多个领域开发创新解决方案。该项目将包括至少两名研究生的培训。该项目的技术方面采用了一种强大的方法，称为有限元外观微积分（FEEC），该方法已经证明有效地分析有限元离散化的稳定性，并揭示了解决各种部分偏微分方程（PDES）的新有限元。目标是将FEEC的范围扩展到设计张量的有限元元素空间，并构建其他有限元络合物。该项目将利用Bernstein-Gelfand-Gelfand（BGG）结构来系统地为各种应用（例如Hessian复合物，弹性复合物和Divdiv Complex）开发有限的元件复合物。此外，该项目旨在使用弱稳定性和分布有限元元素复合物统一不合格的有限元方法，弱的盖尔金元素方法以及虚拟元素方法，从而扩展了有限元元素周期表。 Research topics include smooth finite element spaces for scalar functions, div-conforming face elements for tensors, finite element de Rham and Stokes complexes, discrete BGG construction, weak div and divdiv stability, and distributional finite element complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.