Direct Schur Complement Method by Hierarchical Matrix Techniques

  • Wolfgang Hackbusch
  • Boris N. Khoromskij
  • Ronald Kriemann
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


The goal of this paper is the construction of a data-sparse approximation to the Schur complement on the interface corresponding to FEM and BEM approximations of an elliptic equation by domain decomposition. Using the hierarchical (-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost \(\mathcal{O}\)(N Γ logq N Γ ) is almost linear in N Γ — the number of degrees of freedom on the interface. As input, we use the Schur complement matrices corresponding to subdomains and represented in the -matrix format. In the case of piecewise constant coefficients these matrices can be computed via the BEM representation with the cost \(\mathcal{O}\)(N Γ logq N Γ ), while in the general case the FEM discretisation leads to the complexity O(N Ω logq N Ω ).


Domain Decomposition Linear Complexity Boundary Integral Operator Interface Equation Calderon Projection 


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  1. M. Costabel. Boundary integral operators on lipschitz domains: elementary results. SIAM J. Math. Anal., 19:613–625, 1988.MATHMathSciNetCrossRefGoogle Scholar
  2. L. Grasedyck and W. Hackbusch. Construction and arithmetics of -matrices. Computing, 70:295–334, 2003.MathSciNetCrossRefGoogle Scholar
  3. W. Hackbusch. Integral equations. Theory and numerical treatment, volume 128 of ISNM. Birkhäuser, Basel, 1995.Google Scholar
  4. W. Hackbusch. Direct domain decomposition using the hierarchical matrix technique. In I. Herrera, D. Keyes, O. Widlund, and R. Yates, editors, DDM14 Conference Proceedings, pages 39–50, Mexico City, Mexico, 2003. UNAM.Google Scholar
  5. W. Hackbusch, B. Khoromskij, and R. Kriemann. Hierarchical matrices based on a weak admissibility criterion. Technical Report 2, MPI for Math. in the Sciences, Leipzig, 2003, submitted.Google Scholar
  6. G. Hsiao, B. Khoromskij, and W. Wendland. Preconditioning for boundary element methods in domain decomposition. Engineering Analysis with Boundary Elements, 25:323–338, 2001.CrossRefGoogle Scholar
  7. B. Khoromskij and M. Melenk. Boundary concentrated finite element methods. SIAM J. Numer. Anal., 41:1–36, 2003.MathSciNetCrossRefGoogle Scholar
  8. B. Khoromskij and G. Wittum. Numerical solution of elliptic differential equations by reduction to the interface. Number 36 in LNCSE. Springer, 2004.Google Scholar
  9. U. Langer and O. Steinbach. Boundary element tearing and interconnecting methods. Computing, To appear, 2003.Google Scholar
  10. W. Wendland. Strongly elliptic boundary integral equations. In A. Iserles and M. Powell, editors, The state of the art in numerical analysis, pages 511–561, Oxford, 1987. Clarendon Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Wolfgang Hackbusch
    • 1
  • Boris N. Khoromskij
    • 2
  • Ronald Kriemann
    • 2
  1. 1.Max-Planck-Institute for Mathematics in the Sciences (MPI MIS)Leipzig
  2. 2.MPI MISGermany

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