  We generalize the construction and analysis of auxiliary space
preconditioners to the n-dimensional finite element subcomplex of the de Rham
complex. These preconditioners are based on a generalization of a decomposition
of Sobolev space functions into a regular part and a potential. A discrete
version is easily established using the tools of finite element exterior
calculus. We then discuss the four-dimensional de Rham complex in detail. By
identifying forms in four dimensions (4D) with simple proxies, form operations
are written out in terms of familiar algebraic operations on matrices, vectors,
and scalars. This provides the basis for our implementation of the
preconditioners in 4D. Extensive numerical experiments illustrate their
performance, practical scalability, and parameter robustness, all in accordance
with the theory.
