Enhancing the Periodic Table of Finite Elements

University Of Arizona, Tucson AZ

Investigators

Abstract

The frontiers of science are explored today by elaborate computer-based simulations of real-world phenomena. An essential tool in this endeavor is the finite element method, which has been used over an incredible range of temporal and spatial scales to deepen our understanding of physical behavior. Modern methods rely on a wide variety of finite element "types," each with its own set of challenges and potential benefits. Many of these element types have been classified by the recently developed and widely distributed Periodic Table of Finite Elements. While the table characterizes the most well-known finite elements, some of the included types have never been implemented in practice, due in part to a missing piece of mathematical theory. Further, the mesh element geometries presented in the table do not include a number of key shapes that are used throughout national laboratories and academia, implying that a need exists for a broader characterization of finite element types. This project will address these mathematical gaps in knowledge, thereby promoting best practices in finite elements for computation in a wide range of contexts. In conjunction with collaborators and student researchers, the PI will carry various tasks aimed at enhancing the usefulness and scope of the framework defined by the Periodic Table of Finite Elements. These tasks include the construction of canonical sets of local basis functions for the serendipity family of finite elements, the description of finite element families for key geometries employed in practice but not appearing in the table, and proofs that certain finite element families have minimal dimension in a mathematically precise sense. Proof techniques and basis constructions will involve the use of generalized barycentric functions, Whitney differential form spaces, and gradient bounds for error estimation. The research activities will be accompanied by an outreach program through the Math Teachers' Circle Network and the American Institute of Mathematics.