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.