Title: The Auxiliary Space Preconditioner for the de Rham Complex

Journal Article · · SIAM Journal on Numerical Analysis
DOI: https://doi.org/10.1137/17m1153376 · OSTI ID:1669220
ORCiD logo [1];  [2];  [3]
  1. Portland State Univ., OR (United States)
  2. Johannes Kepler University of Linz (Austria). Inst. of Computational Mathematics
  3. Portland State Univ., OR (United States) ; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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Here 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.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO); USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
1669220
Report Number(s):
LLNL-JRNL-740551; 894730
Journal Information:
SIAM Journal on Numerical Analysis, Journal Name: SIAM Journal on Numerical Analysis Journal Issue: 6 Vol. 56; ISSN 0036-1429
Publisher:
Society for Industrial and Applied MathematicsCopyright Statement
Country of Publication:
United States
Language:
English

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4D
97 MATHEMATICS AND COMPUTING
HX preconditioner
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