Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades.