Finite Element Complexes

这个项目的重点是提高我们解决复杂数学问题的能力，这些问题在工程和物理等领域有实际应用。其目的是创建一个统一的框架，将更好地理解和解决这些问题，使用尖端的计算技术。除了推进科学知识外，该项目还非常重视教育，并通过培养下一代计算数学家来促进多样性。通过开发易于使用的工具和通过各种平台分享知识，该项目旨在造福整个社会，提高我们对复杂系统的理解，并在多个领域开发创新解决方案。该项目将包括培训至少两名研究生。该项目的技术方面采用了一种称为有限元外部演算（FEEC）的强大方法，该方法已被证明在分析有限元离散化的稳定性和揭示求解各种偏微分方程（PDEs）的新有限元方面是有效的。目标是扩展FEEC的范围，以设计张量的有限元空间和构造额外的有限元复合体。该项目将利用Bernstein-Gelfand-Gelfand （BGG）结构系统地开发各种应用的有限元复合体，如Hessian复合体、弹性复合体和div复合体。此外，本项目寻求利用弱稳定性和分布式有限元复合体将非一致性有限元方法、弱伽辽金单元方法和虚拟单元方法统一起来，从而扩展有限元元素周期表。研究课题包括标量函数的光滑有限元空间、张量的分形面元、有限元de Rham和Stokes复合体、离散BGG构造、弱分形和分形稳定性以及分布有限元复合体。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.