Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, $\boldsymbol H(\mathbf{\operatorname{curl}};\Omega)$ and ${{\boldsymbol H}(\operatorname{Div};\Omega)}$. Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain $\Omega$, among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for $\boldsymbol H(\operatorname{Div};\Omega)$ and $\boldsymbol H(\mathbf{\operatorname{curl}};\Omega)$. At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: $\operatorname{div}$, $\operatorname{\mathbf{curl}}$ and $\operatorname{\mathbf{grad}}$ can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar $\boldsymbol H (\operatorname{Div};\Omega)$- and $\boldsymbol H(\mathbf{\operatorname{curl}};\Omega)$-conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.