Numerische Mathematik

, Volume 112, Issue 4, pp 565–600 | Cite as

Domain decomposition based \({\mathcal H}\) -LU preconditioning

Open Access
Article

Abstract

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an \({{\mathcal H}}\) -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based \({{\mathcal H}}\) -matrices, this new approach yields \({\mathcal H}\) -LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an \({\mathcal H}\) -matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based \({{\mathcal H}}\) -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.

Mathematics Subject Classification (2000)

65F05 65F30 65F50 65N55 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Lars Grasedyck
    • 1
  • Ronald Kriemann
    • 1
  • Sabine Le Borne
    • 2
  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Tennessee Technological UniversityCookevilleUSA

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