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Multipatch Space-Time Isogeometric Analysis of Parabolic Diffusion Problems

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Large-Scale Scientific Computing (LSSC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10665))

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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.

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References

  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)

    MATH  Google Scholar 

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

    Book  Google Scholar 

  3. Evans, J., Hughes, T.: Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numer. Math. 123(2), 259–290 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Book  MATH  Google Scholar 

  9. Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)

    Google Scholar 

  10. Langer, U., Moore, S., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution equations. Comput. Methods Appl. Mech. Eng. 306, 342–363 (2016)

    Article  Google Scholar 

  11. Piegl, L., Tiller, W.: The NURBS Book. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-97385-7

    Book  MATH  Google Scholar 

  12. da Veiga, L.B., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)

    Article  MathSciNet  Google Scholar 

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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.

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Authors and Affiliations

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

    U. Langer & M. Neumüller

  2. RICAM, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    U. Langer & I. Toulopoulos

Authors
  1. U. Langer
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  2. M. Neumüller
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  3. I. Toulopoulos
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Corresponding author

Correspondence to U. Langer .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

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Langer, U., Neumüller, M., Toulopoulos, I. (2018). Multipatch Space-Time Isogeometric Analysis of Parabolic Diffusion Problems. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_2

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  • DOI: https://doi.org/10.1007/978-3-319-73441-5_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73440-8

  • Online ISBN: 978-3-319-73441-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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