  In this paper, we consider the extension of the finite element exterior
calculus from elliptic problems, in which the Hodge Laplacian is an appropriate
model problem, to parabolic problems, for which we take the Hodge heat equation
as our model problem. The numerical method we study is a Galerkin method based
on a mixed variational formulation and using as subspaces the same spaces of
finite element differential forms which are used for elliptic problems. We
analyze both the semidiscrete and a fully-discrete numerical scheme.
