  This article reports on the confluence of two streams of research, one
emanating from the fields of numerical analysis and scientific computation, the
other from topology and geometry. In it we consider the numerical
discretization of partial differential equations that are related to
differential complexes so that de Rham cohomology and Hodge theory are key
tools for the continuous problem. After a brief introduction to finite element
methods, the discretization methods we consider, we develop an abstract Hilbert
space framework for analyzing stability and convergence. In this framework, the
differential complex is represented by a complex of Hilbert spaces and
stability is obtained by transferring Hodge theoretic structures from the
continuous level to the discrete. We show stable discretization is achieved if
the finite element spaces satisfy two hypotheses: they form a subcomplex and
there exists a bounded cochain projection from the full complex to the
subcomplex. Next, we consider the most canonical example of the abstract
theory, in which the Hilbert complex is the de Rham complex of a domain in
Euclidean space. We use the Koszul complex to construct two families of finite
element differential forms, show that these can be arranged in subcomplexes of
the de Rham complex in numerous ways, and for each construct a bounded cochain
projection. The abstract theory therefore applies to give the stability and
convergence of finite element approximations of the Hodge Laplacian. Other
applications are considered as well, especially to the equations of elasticity.
Background material is included to make the presentation self-contained for a
variety of readers.
