FEEC

Finite element exterior calculus (FEEC) is a mathematical framework for the design and analysis of finite element methods that respects the differential-geometric structure of partial differential equations. It combines finite element methods with exterior calculus and de Rham cohomology to develop discrete spaces and operators that preserve key geometric and topological properties of the continuous problem, such as exact sequences and conservation laws.

FEEC originated in the early 2000s through the work of Douglas N. Arnold, Richard Falk, and Ragnar Winther, who introduced a systematic approach to construct compatible finite element spaces that form discrete de Rham complexes. The core idea is to approximate differential forms of various degrees by finite element subspaces that fit into a cochain complex with a discrete exterior derivative that commutes with projection onto the discrete spaces. This structure provides stability, convergence guarantees, and accurate representation of physical constraints like divergence-free or curl-free fields.

The mathematical framework centers on the de Rham sequence of differential forms connected by the exterior derivative, and its finite element analogue, a subcomplex of the continuous sequence. Key components include choosing appropriate finite element spaces for k-forms (such as Whitney forms for 1-forms, Nédélec and Raviart-Thomas elements for vector fields, and standard Lagrange elements for 0-forms), and defining a discrete Hodge star or mass matrix that yields compatible inner products. The commuting projections ensure that statistical and topological properties of the continuous problem are mirrored in the discrete setting.

FEEC has been influential in the numerical solution of Maxwell’s equations, incompressible fluid flow, elasticity, and other problems where preserving structure leads to improved stability and accuracy. It provides a unified perspective on mixed methods, error analysis, and mesh design within a geometric framework.