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Scientific Programming
Volume 22, Issue 2, Pages 157-171

Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems

Pierre Jolivet,1,2,3 Frédéric Hecht,2,3 Frédéric Nataf,2,3 and Christophe Prud'homme4

1Laboratoire J. Kuntzmannn, Université J. Fourier, Grenoble Cedex 9, France
2Laboratoire J.-L. Lions, Université P. et M. Curie, Paris, France
3INRIA, ALPINES research team, Rocquencourt, France
4IRMA, Université de Strasbourg, Strasbourg Cedex, France

Copyright © 2014 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Domain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method.