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The auxiliary space preconditioner for the de Rham complex. (English) Zbl 1402.65017

Summary: 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.

MSC:

65F08 Preconditioners for iterative methods
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs

Software:

BoomerAMG
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Full Text: DOI arXiv

References:

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