A parallel fast boundary element method using cyclic graph decompositions


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bebendorf, M., Kriemann, R.: Fast parallel solution of boundary integral equations and related problems. Comp. Vis. Sci. 8, 121–135 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bebendorf, M.: Hierarchical Matrices. Springer, Berlin (2008)

    MATH  Google Scholar 

  4. 4.

    Burnes, J., Hut, P.: A hierarchical \(O(N \log N)\) force calculation algorithm. Nature 324, 446–449 (1986)

    Article  Google Scholar 

  5. 5.

    Colbourn, C.J., Dinitz, J.H.: The CRC Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, London (2007)

    MATH  Google Scholar 

  6. 6.

    Duffy, M.G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19, 1260–1262 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Eppler, K., Harbrecht, H.: Second-order shape optimization using wavelet BEM. Optim. Methods Softw. 21, 135–153 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gallian, J.A.: Graph Labeling. Electron. J. Comb., Dynamic Survey 6 (2013)

  9. 9.

    Grama, A., Kumar, V., Same, A.: Parallel hierarchical solvers and preconditioners for boundary element methods. SIAM J. Sci. Comput. 20, 337–358 (1998)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element methods by panel clustering. Numer. Math. 54, 463–491 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Karypis, G., Kumar, V.: A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Lukáš, D., Postava, K., životský, O.: A shape optimization method for nonlinear axisymmetric magnetostatics using a coupling of finite and boundary elements. Math. Comp. 82, 1721–1731 (2012)

    MATH  Google Scholar 

  13. 13.

    McLean, W., Tran, T.: A preconditioning strategy for boundary element Galerkin methods. Numer. Meth. Partial Differential Equations 13, 283–301 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Of, G.: Fast multipole methods and applications. Lecture Notes in Applied and Computational Mechanics 29, 135–160 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Olstad, B., Manne, F.: Efficient partitioning of sequences. IEEE Trans. Comp. 44, 1322–1325 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    von Petersdorff, T., Stephan, E.: Multigrid solvers and preconditioners for first kind integral equations. Numer. Meth. Partial Differential Equations 8, 443–450 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Rjasanow, S., Steinbach, O.: The Fast Solution of Boundary Integral Equations. Springer, Berlin (2007)

    MATH  Google Scholar 

  18. 18.

    Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187–207 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Rosa, A.: On certain valuations of the vertices of a graph. In Theory of Graphs, International Symposium, Rome, July 1966. Gordon and Breach, pp. 349–355 (1967)

  20. 20.

    Sagan, H.: Space-Filling Curves. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  21. 21.

    Sauter, S., Schwab, C.: Quadrature for hp-Galerkin BEM in \(\mathbb {R}^{3}\). Numer. Math. 78, 211–258 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Berlin (2010)

    Book  Google Scholar 

  23. 23.

    Singer, J.: A theorem in finite projective geometry and some applications to number theory. Trans. AMS 43, 377–385 (1937)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Dalibor Lukáš.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lukáš, D., Kovář, P., Kovářová, T. et al. A parallel fast boundary element method using cyclic graph decompositions. Numer Algor 70, 807–824 (2015). https://doi.org/10.1007/s11075-015-9974-9

Download citation


  • Boundary element method
  • Parallel computing
  • Graph decomposition