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Multipatch Space-Time Isogeometric Analysis of Parabolic Diffusion Problems

  • U. LangerEmail author
  • M. Neumüller
  • I. Toulopoulos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10665)

Abstract

We present and analyze a new stable multi-patch space-time Isogeometric Analyis (IgA) method for the numerical solution of parabolic diffusion problems. The discrete bilinear form is elliptic on the IgA space with respect to a mesh-dependent energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields a priori discretization error estimates. We propose an efficient implementation technique via tensor product representation, and fast space-time parallel solvers. We present numerical results confirming the efficiency of the space-time solvers on massively parallel computers using more than 100.000 cores.

Keywords

Parabolic initial-boundary value problems Space-time isogeometric analysis A priori discretization error estimates Parallel solvers 

Notes

Acknowledgments

The authors gratefully acknowledge the financial support by the Austrian Science Fund (FWF) under the grants NFN S117-03. We also want to thank the Lawrence Livermore National Laboratory for the possibility to perform numerical test on the Vulcan Cluster. In particular, the second author wants to thank P. Vassilevski for the support and the fruitful discussions during his visit at the Lawrence Livermore National Laboratory.

References

  1. 1.
    Bazilevs, Y., Beirão da Veiga, L., Cottrell, J., Hughes, T., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2006)zbMATHGoogle Scholar
  2. 2.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)CrossRefGoogle Scholar
  3. 3.
    Evans, J., Hughes, T.: Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numer. Math. 123(2), 259–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition Methods. CMCS, vol. 9, pp. 69–114. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-23321-5_3. http://www.unige.ch/~gander/Preprints/50YearsTimeParallel.pdf
  5. 5.
    Gander, M., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016).  https://doi.org/10.1137/15M1046605 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hofer, C., Langer, U., Neumüller, M., Toulopoulos, I.: Multipatch time discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems. Under preperation (2017)Google Scholar
  7. 7.
    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, 4135–4195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, New York (1985).  https://doi.org/10.1007/978-1-4757-4317-3 CrossRefzbMATHGoogle Scholar
  9. 9.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)Google Scholar
  10. 10.
    Langer, U., Moore, S., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution equations. Comput. Methods Appl. Mech. Eng. 306, 342–363 (2016)CrossRefGoogle Scholar
  11. 11.
    Piegl, L., Tiller, W.: The NURBS Book. Springer, Heidelberg (1997).  https://doi.org/10.1007/978-3-642-97385-7 CrossRefzbMATHGoogle Scholar
  12. 12.
    da Veiga, L.B., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Computational MathematicsJohannes Kepler University LinzLinzAustria
  2. 2.RICAMAustrian Academy of SciencesLinzAustria

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