Skip to main content
Book cover

Chemnitz Fine Element Symposium

FEM 2017: Advanced Finite Element Methods with Applications pp 187–204Cite as

A Three-Level Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 128)

Abstract

A three-level extension of the GDSW overlapping Schwarz preconditioner in two dimensions is presented, constructed by recursively applying the GDSW preconditioner to the coarse problem. Numerical results, obtained for a parallel implementation using the Trilinos software library, are presented for up to 90,000 cores of the JUQUEEN supercomputer. The superior weak parallel scalability of the three-level method is verified. For large problems and a large number of cores, the three-level method is faster by more than a factor of two, compared to the standard two-level method . The three-level method can also be expected to scale when the classical method will already be out-of-memory.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-14244-5_10
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-14244-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   149.99
Price excludes VAT (USA)
Hardcover Book
USD   149.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 10.1
Fig. 10.2
Fig. 10.3
Fig. 10.4
Fig. 10.5
Fig. 10.6
Fig. 10.7
Fig. 10.8
Fig. 10.9
Fig. 10.10
Fig. 10.11

References

  1. Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)

    CrossRef  MathSciNet  Google Scholar 

  2. Badia, S., Martí n, A.F., Principe, J.: Multilevel balancing domain decomposition at extreme scales. SIAM J. Sci. Comput. 38(1), C22–C52 (2016)

    Google Scholar 

  3. Bavier, E., Hoemmen, M., Rajamanickam, S., Thornquist, H., Amesos2 and Belos: direct and iterative solvers for large sparse linear systems. Sci. Program. 20(3), 241–255 (2012)

    Google Scholar 

  4. Dohrmann, C.R., Klawonn, A., Widlund, O.B.: Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)

    CrossRef  MathSciNet  Google Scholar 

  5. Dohrmann, C.R., Klawonn, A., Widlund, O.B.: A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners. In: Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol. 60, pp. 247–254. Springer, Berlin (2008)

    Google Scholar 

  6. Dohrmann, C.R., Widlund, O.B.: On the design of small coarse spaces for domain decomposition algorithms. SIAM J. Sci. Comput. 39(4), A1466–A1488 (2017)

    CrossRef  MathSciNet  Google Scholar 

  7. Hackbusch, W.: Multigrid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)

    Google Scholar 

  8. Heinlein, A.: Parallel Overlapping Schwarz Preconditioners and Multiscale Discretizations with Applications to Fluid-Structure Interaction and Highly Heterogeneous Problems. PhD thesis, Universität zu Köln (2016)

    Google Scholar 

  9. Heinlein, A., Klawonn, A., Rheinbach, O.: A parallel implementation of a two-level overlapping Schwarz method with energy-minimizing coarse space based on Trilinos. SIAM J. Sci. Comput. 38(6), C713–C747 (2016)

    CrossRef  MathSciNet  Google Scholar 

  10. Heinlein, A., Klawonn, A., Rheinbach, O.: Parallel Two-Level Overlapping Schwarz Methods in Fluid-Structure Interaction, pp. 521–530. Springer International Publishing, Cham (2016)

    Google Scholar 

  11. Heinlein, A., Klawonn, A., Rheinbach, O., Widlund, O.: Improving the parallel performance of overlapping Schwarz methods by using a smaller energy minimizing coarse space. 2017. Accepted for publication in the Proceedings of the 24rd International Conference on Domain Decomposition Methods. Springer Lecture Notes in Computational Science and Engineering

    Google Scholar 

  12. Heinlein, A., Klawonn, A., Rajamanickam, S., Rheinbach, O.: FROSch: a fast and robust overlapping Schwarz domain decomposition preconditioner based on Xpetra in Trilinos. Technischer Bericht, Universität zu Köln, November 2018. https://kups.ub.uni-koeln.de/9018/

    CrossRef  Google Scholar 

  13. Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos Project. ACM Trans. Math. Softw. 31(3), 397–423 (2005)

    CrossRef  MathSciNet  Google Scholar 

  14. Kong, F., Cai, X.-C.: A highly scalable multilevel Schwarz method with boundary geometry preserving coarse spaces for 3D elasticity problems on domains with complex geometry. SIAM J. Sci. Comput. 38(2), C73–C95 (2016)

    CrossRef  MathSciNet  Google Scholar 

  15. Mandel, J., Sousedík, B., Dohrmann, C.R.: Multispace and multilevel BDDC. Computing 83(2–3), 55–85 (2008)

    CrossRef  MathSciNet  Google Scholar 

  16. Rajamanickam, S., Boman, E.G., Heroux, M.A.: Shylu: a hybrid-hybrid solver for multicore platforms. In: 2012 IEEE 26th International Parallel and Distributed Processing Symposium, May 2012, pp. 631–643 (2012)

    Google Scholar 

  17. Sala, M., Stanley, K.S, Heroux, M.A: On the design of interfaces to sparse direct solvers. ACM Trans. Math. Softw. 34(2), 9:1–9:22 (2008)

    Google Scholar 

  18. Scacchi, S.: A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197(45–48), 4051–4061 (2008)

    CrossRef  MathSciNet  Google Scholar 

  19. Stephan, M., Docter, J.: JUQUEEN: IBM Blue Gene/Q® Supercomputer System at the Julich Supercomputing Centre. Journal of Large-Scale Research Facilities 1, A1 (2015)

    CrossRef  Google Scholar 

  20. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

    Google Scholar 

  21. Trilinos public git repository. Web, 2018. https://github.com/trilinos/trilinos

  22. Tu, X.: Three-level BDDC in two dimensions. Int. J. Numer. Methods Eng. 69(1), 33–59 (2007)

    CrossRef  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the German Research Foundation (DFG) through the Priority Programme 1648 “Software for Exascale Computing” (SPPEXA) under grants RH 122/3-2 and KL 2094/4-2.

Also, this work is in part supported under grant RH 122/5-2.

The authors gratefully acknowledge the computing the Gauss Centre for Supercomputing e. V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer JUQUEEN BG/Q supercomputer [19] at JSC Jülich. GCS is the alliance of the three national supercomputing centres HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayrische Akademie der Wissenschaften).

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Köln, Germany

    Alexander Heinlein & Axel Klawonn

  2. Institut für Numerische Mathematik und Optimierung, Technische Universität Bergakademie Freiberg, Freiberg, Germany

    Oliver Rheinbach & Friederike Röver

Authors
  1. Alexander Heinlein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Axel Klawonn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oliver Rheinbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Friederike Röver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver Rheinbach .

Editor information

Editors and Affiliations

  1. Institut für Mathematik & Computergestützte Simulation, Universität der Bundeswehr München, Neubiberg, Germany

    Prof. Dr. Thomas Apel

  2. Institute for Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Prof. Dr. Ulrich Langer

  3. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Prof. Dr. Arnd Meyer

  4. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria

    Prof. Dr. Olaf Steinbach

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Heinlein, A., Klawonn, A., Rheinbach, O., Röver, F. (2019). A Three-Level Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_10

Download citation