  A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that
a large class of mixed finite element methods can be formulated naturally on
Hilbert complexes, where using a Galerkin-like approach, one solves a
variational problem on a finite-dimensional subcomplex. In a seemingly
unrelated research direction, Dziuk [Lect Notes in Math, vol 1357 (1988)]
analyzed a class of nodal finite elements for the Laplace-Beltrami equation on
smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow
later extended this analysis [SIAM J Numer Anal, 47 (2009)] to 3-surfaces, as
well as to higher-order surface approximation. In this article, we bring these
lines of research together, first developing a framework for the analysis of
variational crimes in abstract Hilbert complexes, and then applying this
abstract framework to the setting of finite element exterior calculus on
hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to
problems that violate their subcomplex assumption, allowing for the extension
of finite element exterior calculus to approximate domains, most notably the
Hodge-de Rham complex on approximate manifolds. As an application of the
latter, we recover Dziuk's and Demlow's a priori estimates for 2- and
3-surfaces, demonstrating that surface finite element methods can be analyzed
completely within this abstract framework. Moreover, our results generalize
these earlier estimates dramatically, extending them from nodal finite elements
for Laplace-Beltrami to mixed finite elements for the Hodge Laplacian, and from
2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By
developing this analytical framework using a combination of general tools from
differential geometry and functional analysis, we are led to a more geometric
analysis of surface finite element methods, whereby the main results become
more transparent.
