Spectral Analysis of Geometric Multigrid Methods for Isogeometric Analysis

  • Clemens Hofreither
  • Walter Zulehner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8962)


We investigate geometric multigrid methods for solving the large, sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. We observe that the performance of standard V-cycle iteration is highly dependent on the spatial dimension as well as the spline degree of the discretization space. Conjugate gradient iteration preconditioned with one V-cycle mitigates this dependence, but does not eliminate it. We perform both classical local Fourier analysis as well as a numerical spectral analysis of the two-grid method to gain better understanding of the underlying problems and observe that classical smoothers do not perform well in the isogeometric setting.



This work was supported by the National Research Network “Geometry + Simulation” (NFN S117, 2012–2016), funded by the Austrian Science Fund (FWF). The first author was also supported by the project AComIn “Advanced Computing for Innovation”, grant 316087, funded by the FP7 Capacity Programme “Research Potential of Convergence Regions”.


  1. 1.
    Brandt, A.: Rigorous quantitative analysis of multigrid, I: Constant coefficients two-level cycle with \(L_2\)-norm. SIAM J. Numer. Anal. 31(6), 1695–1730 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Buffa, A., Harbrecht, H., Kunoth, A., Sangalli, G.: BPX-preconditioning for isogeometric analysis. Comput. Methods. Appl. Mech. Eng. 265, 63–70 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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). CrossRefMathSciNetGoogle Scholar
  4. 4.
    Gahalaut, K.P.S., Kraus, J.K., Tomar, S.K.: Multigrid methods for isogeometric discretization. Computer Methods in Applied Mechanics and Engineering 253, 413–425 (2013). CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    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). CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kleiss, S.K., Pechstein, C., Jüttler, B., Tomar, S.: IETI - isogeometric tearing and interconnecting. Comput. Methods Appl. Mech. Eng. 247–248, 201–215 (2012)CrossRefGoogle Scholar
  7. 7.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, San Diego (2001)zbMATHGoogle Scholar
  8. 8.
    Beirão da Veiga, L., Cho, D., Pavarino, L., Scacchi, S.: Overlapping Schwarz methods for isogeometric analysis. SIAM J. Numer. Anal. 50(3), 1394–1416 (2012). CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Computational MathematicsJohannes Kepler University LinzLinzAustria

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