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.