Parallelization Strategies for Matrix Assembly in Finite Element Methods

  • Jorn Behrens


Adaptive finite element methods include unstructured discretization meshes. Most algorithms for the matrix assembly follow the ordering of the arising meshes. Time-dependent problems resulting in rapidly changing locally refined (i.e. unstructured) discretization grids need a matrix assembly in each timestep. This demands a considerable amount of computing time which in turn calls for parallelization.

Domain decomposition techniques often suffer from insufficient load balancing. Especially in adaptive methods balancing the computational workload is difficult, because the number of unknowns in each domain is not predictable and even may change during the computation.

Analysing data dependencies in the matrix assembly, one obtains the problem to synchronize access to one node (matrix element) from neighbouring elements. An efficient algorithm based on coloring the elements following a very simple tabulated indexing scheme has been found. Within each color, calculation is free of data dependencies. Load balancing becomes easy, because data can be distributed uniformly among the processors. This method works well for shared memory architectures.

For distributed memory computers, however, the color indexing method fails, because logical and physical ordering differ, thus physical data access is not structured. A second method for parallelizing the same problem (i.e. matrix assembly in finite element methods) has been developed. Using index-arrays for mapping the logical ordering to a suitable physical (storage-) ordering, one obtains an almost optimally load balanced and efficient algorithm.

Timing results for an Alliant FX/2800 (color indexing) and a Kendall Square Research KSR-1 show the efficiency of both algorithms.

Timing results for an Alliant FX/2800 (color indexing) and a Kendall Square Research KSR-1 show the efficiency of both algorithms.


Color Indexing Matrix Assembly Memory Architecture Parallelization Strategy Local Refinement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Alliant Computer Systems Corp., Littleton (Massachusetts). FX/2800 System Description, mar 1991. Part no. 300–00500.Google Scholar
  2. [2]
    J. Behrens. Optimierung eines Mehrgitterverfahrens und eines Hierarchische Basen Verfahrens auf einer Alliant FX/80. Diplomarbeit, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1990. Erstellt am Alfred-Wegener-Institut Bremerhaven.Google Scholar
  3. [3]
    W. Hiller and J. Behrens. Parallelisierung von Mehrgitteralgorithmen auf der Alliant FX/80. In H. W. Meuer, editor, Parallelisierung komplexer Probleme, pages 37–82, Berlin, 1991. Springer-Verlag.CrossRefGoogle Scholar
  4. [4]
    R. W. Hockney and C. R. Jesshope. Parallel Computers 2, Architecture, Programming and Algorithms. Adam Hilger, Bristol Philadelphia, 2nd edition, 1988.Google Scholar
  5. [5]
    M. T. Jones and P. E Plassmann. A parallel graph coloring heuristic. SIAM J. Sci. Comput., 14 (3): 654–669, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Kendall Square Research Corp., Waltham. KSR1 System Administration, Dec. 1992.Google Scholar
  7. [7]
    G. Lang. Kurzbeschreibung des Programmes PFRG02. Technical report, Bundesanstalt für Wasserbau, Hamburg, 1992.Google Scholar
  8. [8]
    J. Liou and T. E. Tezduyar. Clustered element-by-element computations for fluid flow. In H. D. Simon, editor, Parallel Computational Fluid Dynamics: Implementations and Results, Scientific and Engineering Computation, pages 167–187, Cambridge Massachusetts, 1992. The MIT Press.Google Scholar
  9. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. in Math. Softw,15(4):326–347, 1989.zbMATHCrossRefGoogle Scholar
  10. [10]
    H. Yserentant. On the multi-level splitting of finite element spaces. Numer. Math., 49: 379–412, 1986.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Jorn Behrens
    • 1
  1. 1.Institute for Polar and Marine ResearchAlfred-Wegener-InstitutBremerhavenGermany

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