New matrix-by-vector multiplications based on a nonoverlapping domain decomposition data distribution

  • Gundolf Haase
Workshop 09: Parallel Numerical Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1300)


The nonoverlapping domain decomposition (DD) method allows to store vectors as accumulated or as distributed (each process stores a part of that value vectors. The case of distributed stored matrices is fully investigated, whereas the use of accumulated matrices is not customary in the DD community. This paper is concerned with the accumulated matrices and investigates conditions under which matrix-by-vector multiplications can be realized. Especially, we derive restrictions on the matrix shape and the finite element (FE) mesh. As a new result, the special structure of these admissible accumulated matrices leads directly to global incomplete factorizations as preconditioners in CG-like methods. Also, the well-known DD preconditioners fit into the general framework of the matrix-by-vector operations presented.


Parallel Iterative Solvers Incomplete Factorization Preconditioning Domain Decomposition Finite Element Method 


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  1. 1.
    J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring I–IV. Mathematics of Computation, 1986, 1987, 1988, 1989. 47, 103–134, 49, 1-16, 51, 415-430, 53, 1-24.Google Scholar
  2. 2.
    U. Groh. Local realization of vector operations on parallel computers. Preprint SPC 94-2, TU Chemnitz, 1994. in german.Google Scholar
  3. 3.
    I. Gustafsson. A class of first order factorization methods. BIT, 18:142–156, 1978.Google Scholar
  4. 4.
    G. Haase. Hierarchical extension operators plus smoothing in domain decomposition preconditioners. Applied Numerical Mathematics, 23(3):327–346, May 1997.Google Scholar
  5. 5.
    G. Haase, U. Langer, and A. Meyer. The approximate Dirichlet decomposition method. part I,II. Computing, 47:137–167, 1991.Google Scholar
  6. 6.
    M. Jung. On the parallelization of multi-grid methods using a non-overlapping domain decomposition data structure. Applied Numerical Mathematics, 23(1), 1997.Google Scholar
  7. 7.
    K. H. Law. A parallel finite element solution method. Computer and Structures, 23(6):845–858, 1989.Google Scholar
  8. 8.
    A. Meyer. A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain. Computing, 45:217–234, 1990.Google Scholar
  9. 9.
    B. Smith, P. B. rstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996.Google Scholar
  10. 10.
    C. H. Tong, T. F. Chan, and C. J. Kuo. Multilevel filtering preconditioners: Extensions to more general elliptic problems. SIAM J. Sci. Stat. Comput., 13:227–242, 1992. *** DIRECT SUPPORT *** A0008C42 00025Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Gundolf Haase
    • 1
  1. 1.University LinzInst. of Math.LinzAustria

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