This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was also supported in part by the AFOSR under grant FA9550-17-1-0090 and by the ARO under US Army Federal Grant W911NF-15-1-0590.
Finite element method, Inequalities (Mathematics)
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.
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Gopalakrishnan, J., Neumüller, M., and Vassilevski, P. S., (2018). The Auxiliary Space Preconditioner for the de Rham Complex. SIAM J. Numer. Anal., 56(6), 3196–3218.