  Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been
developed as a general framework for linear mixed variational problems, their
numerical approximation by mixed methods, and their error analysis. The basic
approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal
articles in 2006 and 2010, interprets these problems in the setting of Hilbert
complexes, leading to a more general and complete understanding. Over the last
five years, the FEEC framework has been extended to a broader set of problems.
One such extension, due to Holst and Stern in 2012, was to problems with
variational crimes, allowing for the analysis and numerical approximation of
linear and geometric elliptic partial differential equations on Riemannian
manifolds of arbitrary spatial dimension. Their results substantially
generalize the existing surface finite element approximation theory in several
respects. In 2014, Gillette, Holst, and Zhu extended the FEEC in another
direction, namely to parabolic and hyperbolic evolution systems by combining
the FEEC framework for elliptic operators with classical approaches for
parabolic and hyperbolic operators, by viewing solutions to the evolution
problem as lying in Bochner spaces (spaces of Banach-space valued parametrized
curves). Related work on developing an FEEC theory for parabolic evolution
problems has also been done independently by Arnold and Chen. In this article,
we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems
on Riemannian manifolds, through the use of framework developed by Holst and
Stern for analyzing variational crimes. We establish a priori error estimates
that reduce to the results from earlier work in the flat (non-criminal)
setting. Some numerical examples are also presented.
