Numerische Mathematik

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

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

  • Lars GrasedyckEmail author
  • Ronald Kriemann
  • Sabine Le Borne
Open Access


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 


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.


  1. 1.
    Bebendorf M.: Why approximate LU decompositions of finite element discretizations of elliptic operators can be computed with almost linear complexity. SIAM J. Numer. Anal. 45, 1472–1494 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bebendorf M., Hackbusch W.: Existence of \({{\mathcal{H}}}\) -matrix approximants to the inverse FE-matrix of elliptic operators with L -coefficients. Numer. Math. 95, 1–28 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Börm S.: \({{\mathcal H}^{2}}\) -matrix arithmetics in linear complexity. Computing 77, 1–28 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brainman I., Toledo S.: Nested-dissection orderings for sparse LU with partial pivoting. SIAM J. Math. Anal. Appl. 23, 998–1012 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gavrilyuk I., Hackbusch W., Khoromskij B.: \({\mathcal{H}}\) -matrix approximation for the operator exponential with applications. Numer. Math. 92, 83–111 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    George A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345–363 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. Ph.D. thesis, Universität Kiel (2001)Google Scholar
  8. 8.
    Grasedyck L., Le Borne S.: \({\mathcal H}\) -matrix preconditioners in convection-dominated problems. SIAM J. Math. Anal. 27, 1172–1183 (2005)MathSciNetGoogle Scholar
  9. 9.
    Grasedyck L., Hackbusch W.: Construction and arithmetics of \({{\mathcal{H}}}\) -matrices. Computing 70, 295–334 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grasedyck L., Hackbusch W., Kriemann R.: Performance of \({\mathcal{H}}\) -LU preconditioning for sparse matrices. Comput. Methods Appl. Math. 8, 336–349 (2008)zbMATHGoogle Scholar
  11. 11.
    Grasedyck L., Kriemann R., Le Borne S.: Parallel black box \({\mathcal{H}}\) -LU preconditioning for elliptic boundary value problems. Comput. Visual. Sci. 11(4–6), 273–291 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hackbusch W.: A sparse matrix arithmetic based on \({\mathcal{H}}\) -matrices. Part I: Introduction to \({\mathcal{H}}\) -matrices. Computing 62, 89–108 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hackbusch, W.: Direct domain decomposition using the hierarchical matrix technique. In: Herrera, I., Keyes, D., Widlund, O., Yates, R. (eds.) Domain Decomposition Methods in Science and Engineering, UNAM, pp. 39–50 (2003)Google Scholar
  14. 14.
    Hackbusch W., Khoromskij B.: \({\mathcal{H}}\) -matrix approximation on graded meshes. In: Whiteman, J.R. (eds) The Mathematics of Finite Elements and Applications, pp. 307–316. Elsevier, Amsterdam (2000)CrossRefGoogle Scholar
  15. 15.
    Hackbusch W., Khoromskij B.: A sparse matrix arithmetic based on \({\mathcal{H}}\) -matrices. Part II: Application to multi-dimensional problems. Computing 64, 21–47 (2000)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Hackbusch W., Khoromskij B., Sauter S.: On \({\mathcal{H}^{2}}\) -matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds) Lectures on Applied Mathematics, pp. 9–29. Springer, Berlin (2000)Google Scholar
  17. 17.
    Hendrickson B., Rothberg E.: Improving the run time and quality of nested dissection ordering. SIAM J. Sci. Comp. 20, 468–489 (1998)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Ibragimov I., Rjasanow S., Straube K.: Hierarchical Cholesky decomposition of sparse matrices arising from curl-curl-equations. J. Numer. Math. 15, 31–58 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Le Borne, S.: Hierarchical matrices for convection-dominated problems. In: Kornhuber, R., Hoppe, R., Périaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, pp. 631–638 (2004)Google Scholar
  20. 20.
    Le Borne S., Grasedyck L., Kriemann R.: Domain-decomposition based H-LU preconditioners. In: Widlund, O.B., Keyes, D.E. (eds) Domain Decomposition Methods in Science and Engineering XVI. Lecture Notes in Computational Science and Engineering, vol. 55, pp. 661–668. Springer, Berlin (2006)Google Scholar
  21. 21.
    Lipton R.J., Rose D.J., Tarjan R.E.: Generalized nested dissection. SIAM J. Numer. Anal. 16, 346–358 (1979)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

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

Personalised recommendations