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 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 four dimensions. Extensive numerical experiments illustrate their performance, practical scalability, and parameter robustness, all in accordance with the theory.


  1. regular decomposition
  2. HX preconditioner
  3. 4D
  4. skew-symmetric matrix fields
  5. exterior derivative
  6. proxies

MSC codes

  1. 65F08
  2. 65N30

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 3196 - 3218
ISSN (online): 1095-7170


Submitted: 23 October 2017
Accepted: 29 August 2018
Published online: 6 November 2018


  1. regular decomposition
  2. HX preconditioner
  3. 4D
  4. skew-symmetric matrix fields
  5. exterior derivative
  6. proxies

MSC codes

  1. 65F08
  2. 65N30



Funding Information

Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-17-1-0090
Army Research Office https://doi.org/10.13039/100000183 : W911NF-15-1-0590, W911NF-16-1-0307
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1624776
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-AC52-07NA27344

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