Skip to main content
SpringerLink
Log in

Mass Smoothers in Geometric Multigrid for Isogeometric Analysis

  • Conference paper
  • First Online:
Curves and Surfaces (Curves and Surfaces 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

Included in the following conference series:

Abstract

We investigate geometric multigrid methods for solving the large, sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. In particular, we study a smoother which incorporates the inverse of the mass matrix as an iteration matrix, and which we call mass-Richardson smoother. We perform a rigorous analysis in a model setting and perform some numerical experiments to confirm the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(07), 1031–1090 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for \(h\)-\(p\)-\(k\)-refinement in isogeometric analysis. Numer. Math. 118(2), 271–305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buffa, A., Harbrecht, H., Kunoth, A., Sangalli, G.: BPX-preconditioning for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 265, 63–70 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Collier, N., Pardo, D., Dalcin, L., Paszynski, M., Calo, V.M.: The cost of continuity: a study of the performance of isogeometric finite elements using direct solvers. Comput. Methods Appl. Mech. Eng. 213–216, 353–361 (2012)

    Article  MathSciNet  Google Scholar 

  5. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)

    Book  Google Scholar 

  6. Beirão da Veiga, L., Cho, D., Pavarino, L., Scacchi, S.: Overlapping Schwarz methods for isogeometric analysis. SIAM J. Num. Anal. 50(3), 1394–1416 (2012)

    Article  MATH  Google Scholar 

  7. Beirão da Veiga, L., Cho, D., Pavarino, L., Scacchi, S.: BDDC preconditioners for isogeometric analysis. Math. Models Methods Appl. Sci. 23(6), 1099–1142 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beirão da Veiga, L., Pavarino, L., Scacchi, S., Widlund, O., Zampini, S.: Isogeometric BDDC preconditioners with deluxe scaling. SIAM J. Sci. Comput. 36(3), A1118–A1139 (2014)

    Article  MATH  Google Scholar 

  9. Gahalaut, K.P.S., Kraus, J.K., Tomar, S.K.: Multigrid methods for isogeometric discretization. Comput. Methods Appl. Mech. Eng. 253, 413–425 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, L., Calo, V.M.: Fast isogeometric solvers for explicit dynamics. Comput. Methods Appl. Mech. Eng. 274, 19–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hackbusch, W.: Multi-Grid Methods and Applications, vol. 4. Springer Series in Computational Mathematics. Springer-Verlag, Heidelberg (2003)

    Google Scholar 

  12. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kleiss, S.K., Pechstein, C., Jüttler, B., Tomar, S.: IETI - Isogeometric tearing and interconnecting. Comput. Methods Appl. Mech. Eng. 247–248, 201–215 (2012)

    Article  Google Scholar 

  14. Schwab, C.: \(p\)- and \(hp\)-Finite Element Methods. Clarendon Press, Oxford (1998)

    Google Scholar 

Download references

Acknowledgments

This work was supported by the National Research Network “Geometry + Simulation” (NFN S117, 2012–2016), funded by the Austrian Science Fund (FWF).

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University Linz, Altenbergerstr. 69, 4040, Linz, Austria

    Clemens Hofreither & Walter Zulehner

Authors
  1. Clemens Hofreither
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Zulehner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clemens Hofreither .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Hofreither, C., Zulehner, W. (2015). Mass Smoothers in Geometric Multigrid for Isogeometric Analysis. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-22804-4_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation