  Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed
that mixed variational problems, and their numerical approximation by mixed
methods, could be most completely understood using the ideas and tools of
Hilbert complexes. This led to the development of the Finite Element Exterior
Calculus (FEEC) for a large class of linear elliptic problems. More recently,
Holst and Stern [Found. Comp. Math. 12:3 (2012), 263--293 and 363--387]
extended the FEEC framework to semi-linear problems, and to problems containing
variational crimes, allowing for the analysis and numerical approximation of
linear and nonlinear geometric elliptic partial differential equations on
Riemannian manifolds of arbitrary spatial dimension, generalizing surface
finite element approximation theory. In this article, we develop another
distinct extension to the FEEC, namely to parabolic and hyperbolic evolution
systems, allowing for the treatment of geometric and other evolution problems.
Our approach is to combine the recent work on the FEEC for elliptic problems
with a classical approach to solving evolution problems via semi-discrete
finite element methods, by viewing solutions to the evolution problem as lying
in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical
approaches by Thomee for parabolic problems and Geveci for hyperbolic problems,
we establish a priori error estimates for Galerkin FEM approximation in the
natural parametrized Hilbert space norms. In particular, we recover the results
of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods
as special cases, effectively extending their results to arbitrary spatial
dimension and to an entire family of mixed methods. We also show how the Holst
and Stern framework allows for extensions of these results to certain
semi-linear evolution problems.
