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

The auxiliary space preconditioner for the de Rham complex