Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl; Omega) and H(Div; Omega). Thus finite element subspaces of these function s paces are indispensable for effective finite element discretization schemes . Given a simplicial triangulation of the computational domain Omega, among o thers, Raviart, Thomas and Nedelec have found suitable conforming finite el ements for H(Div; Omega) and H(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 b etween differential forms and differential operators: div, curl and grad ca n all be regarded as special incarnations of the exterior derivative of a d ifferential form. Moreover, in the realm of differential forms most concept s are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming fin ite 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 elem ent spaces. With unprecedented ease we can recover the familiar H(Div; Omeg a)- and H(curl; Omega)-conforming finite elements, and establish the unisol vence of degrees of freedom. In addition, the use of differential forms mak es it possible to establish crucial algebraic properties of the canonical i nterpolation operators and representation theorems in a single sweep for al l kinds of spaces.