Numerical Algorithms

, Volume 70, Issue 4, pp 807–824 | Cite as

A parallel fast boundary element method using cyclic graph decompositions

  • Dalibor LukášEmail author
  • Petr Kovář
  • Tereza Kovářová
  • Michal Merta
Original Paper


We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is \(O((n/\sqrt {N})\log (n/\sqrt {N}))\) in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n=2744832 and N=273 on a real-world geometry document that the method exhibits superior parallel scalability \(O((n/N)\,\log n)\) of the overall time, while the memory consumption scales accordingly to the theoretical estimate.


Boundary element method Parallel computing Graph decomposition 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Dalibor Lukáš
    • 1
    Email author
  • Petr Kovář
    • 1
  • Tereza Kovářová
    • 1
  • Michal Merta
    • 1
  1. 1.VŠB–Technical University of OstravaOstrava-PorubaCzech Republic

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