Finite Element Tensor Calculus

Abstract

Discrete structures lie at the foundation of modern scientific computation. In electromagnetism, fluid dynamics, and continuum mechanics, the governing laws are expressed through geometric and topological structures rather than mere functions, and preserving these structures is essential for reliable computation.

This lecture series develops mixed finite element methods from the viewpoint of differential complexes. We begin with inf-sup stability and the Babuška-Brezzi theory, and explain how differential complexes provide a unifying framework for the analysis and discretization of partial differential equations. Applications include electromagnetism, fluid mechanics, elasticity and generalized continua, as well as metric and curvature-related problems, spectra, and pseudospectra.

The second part of the course turns to discrete structures. We will discuss continuous and discrete differential forms, the BGG machinery, the Finite Element Periodic Table, and the Hodge Laplacian, and then move on to graphs and networks, Lie transport, and applications to topological hydrodynamics and magnetohydrodynamics. Finite Element Exterior Calculus (FEEC) and Finite Element Tensor Calculus (FETC) provide the main conceptual framework, connecting geometric structure with stable numerical discretization.