## Mathematical Sciences Colloquium

The purpose of the colloquium series is to discuss research problems in pure and applied mathematics, operations research, STEM education, and related topics.

For more information, or if you would like to give a talk, please contact Dr. Pei Liu at pliu@fit.edu

### 2023-2024 Colloquium Series

**The Combinatorics of Finite Element Methods**

**Yakov Berchenko-Kogan, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology**

Monday, August 28, 2023

**Abstract: **

Finite element methods are a way of discretizing and numerically solving partial differential equations. Unfortunately, finite element methods can sometimes give wrong answers, even for simple problems. For example, they can give so-called "spurious modes", which are wrong solutions to an eigenvalue problem. Nonetheless, in the 1970s and 1980s, numerical analysts working in electromagnetism developed new finite element methods that avoided these issues. But, unbeknownst to them, they were actually reinventing spaces developed decades earlier by pure mathematicians working in geometry. Why were geometers working with finite element spaces?

In the 1500s, mathematicians noticed that, for any polyhedron, if one takes the number of vertices, subtracts the number of edges, and adds the number of faces, one always gets two. Meanwhile, in calculus, we learn that any divergence-free vector field B is equal to the curl of another vector field A. We also learn that this fact is false if our domain has a hole: the electric field of a point charge, defined everywhere except at the origin, is divergence-free but is not equal to the curl of any other vector field. By the 1930s, geometers understood that these seemingly unrelated facts are actually directly connected via what is now called de Rham's Theorem, a powerful link between the discrete world and the continuous world using the language of cohomology. Finite element methods are also, at their core, a link between the discrete world of computation and the continuous world of differential equations, and so, by the late 1980s, the fields of pure math and applied math ended up colliding.

Over the next few decades, the connection between finite element theory and differential geometry was more fully understood. In this talk, I will discuss the work of Arnold, Falk, and Winther in 2006-2010, who formalized this connection and explained why some finite element methods work well and others do not: Finite element spaces that respect de Rham's theorem will give correct answers; finite element spaces that do not respect de Rham's theorem might not.