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Functional Type Error Control for Stabilised Space-Time IgA Approximations to Parabolic 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

The paper is concerned with reliable space-time IgA schemes for parabolic initial-boundary value problems. We deduce a posteriori error estimates and investigate their applicability to space-time IgA approximations. Since the derivation is based on purely functional arguments, the estimates do not contain mesh dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they imply estimates for discrete norms associated with stabilised space-time IgA approximations. Finally, we illustrate the reliability and efficiency of presented error estimates for the approximate solutions recovered with IgA techniques on a model example.

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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(7), 1031–1090 (2006). http://dx.doi.org/10.1142/S0218202506001455

    Article  MathSciNet  MATH  Google Scholar 

  2. Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Technical report arXiv:1502.00565 [math.NA] (2015)

  3. Dedè, L., Santos, H.A.F.A.: B-spline goal-oriented error estimators for geometrically nonlinear rods. Comput. Mech. 49(1), 35–52 (2012). http://dx.doi.org/10.1007/s00466-011-0625-2

    Article  MathSciNet  MATH  Google Scholar 

  4. Dörfel, M.R., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local \(h\)-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 264–275 (2010). http://dx.doi.org/10.1016/j.cma.2008.07.012

    Article  MathSciNet  MATH  Google Scholar 

  5. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gaevskaya, A.V., Repin, S.I.: A posteriori error estimates for approximate solutions of linear parabolic problems. Differ. Equ. 41(7), 970–983 (2005). Springer

    Article  MathSciNet  MATH  Google Scholar 

  7. Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29(7), 485–498 (2012). http://dx.doi.org/10.1016/j.cagd.2012.03.025

    Article  MathSciNet  MATH  Google Scholar 

  8. Hansbo, P.: Space-time oriented streamline diffusion methods for nonlinear conservation laws in one dimension. Comm. Numer. Meth. Eng. 10(3), 203–215 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Johannessen, K.: An adaptive isogeometric finite element analysis. Technical report, Master’s thesis, Norwegian University of Science and Technology (2009)

    Google Scholar 

  10. Johnson, C., Saranen, J.: Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comput. 47(175), 1–18 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kleiss, S.K., Tomar, S.K.: Guaranteed and sharp a posteriori error estimates in isogeometric analysis. Comput. Math. Appl. 70(3), 167–190 (2015). http://dx.doi.org/10.1016/j.camwa.2015.04.011

    Article  MathSciNet  MATH  Google Scholar 

  12. Kraft, R.: Adaptive and linearly independent multilevel \(B\)-splines. In: Surface Fitting and Multiresolution Methods (Chamonix-Mont-Blanc, 1996), pp. 209–218. Vanderbilt University Press, Nashville (1997)

    Google Scholar 

  13. Kumar, M., Kvamsdal, T., Johannessen, K.A.: Simple a posteriori error estimators in adaptive isogeometric analysis. Comput. Math. Appl. 70(7), 1555–1582 (2015). http://dx.doi.org/10.1016/j.camwa.2015.05.031

    Article  MathSciNet  Google Scholar 

  14. Kuru, G., Verhoosel, C.V., van der Zee, K.G., van Brummelen, E.H.: Goal-adaptive isogeometric analysis with hierarchical splines. Comput. Methods Appl. Mech. Eng. 270, 270–292 (2014). https://doi.org/10.1016/j.cma.2013.11.026

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Langer, U., Matculevich, S., Repin, S.: A posteriori error estimates for space-time IgA approximations to parabolic initial boundary value problems. arXiv.org/1612.08998 [math.NA] (2016)

  17. 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 

  18. Mali, O., Neittaanmäki, P., Repin, S.: Accuracy Verification Methods. Computational Methods in Applied Sciences, vol. 32. Springer, Dordrecht (2014). https://doi.org/10.1007/978-94-007-7581-7

    MATH  Google Scholar 

  19. Mantzaflaris, A., et al.: G+Smo (geometry plus simulation modules) v0.8.1 (2015). http://gs.jku.at/gismo

  20. Matculevich, S.: Fully reliable a posteriori error control for evolutionary problems. Ph.D. thesis, Jyväskylä Studies in Computing, University of Jyväskylä (2015)

    Google Scholar 

  21. Matculevich, S., Repin, S.: Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation. Appl. Math. Comput. 247, 329–347 (2014). http://dx.doi.org/10.1016/j.amc.2014.08.055

    MathSciNet  MATH  Google Scholar 

  22. Repin, S.: A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter GmbH & Co. KG, Berlin (2008)

    Book  MATH  Google Scholar 

  23. Repin, S.I.: Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation. Rend. Mat. Acc. Lincei 13(9), 121–133 (2002)

    MathSciNet  MATH  Google Scholar 

  24. Repin, S.: A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauchnych Seminarov POMIs 243, 201–214 (1997)

    MATH  Google Scholar 

  25. Tagliabue, A., Dedè, L., Quarteroni, A.: Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput. Fluids 102, 277–303 (2014)

    Article  MathSciNet  Google Scholar 

  26. Vuong, A.V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200(49–52), 3554–3567 (2011). http://dx.doi.org/10.1016/j.cma.2011.09.004

    Article  MathSciNet  MATH  Google Scholar 

  27. van der Zee, K.G., Verhoosel, C.V.: Isogeometric analysis-based goal-oriented error estimation for free-boundary problems. Finite Elem. Anal. Des. 47(6), 600–609 (2011). http://dx.doi.org/10.1016/j.finel.2010.12.013

    Article  MathSciNet  Google Scholar 

  28. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A. Springer, New York (1990). https://doi.org/10.1007/978-1-4612-0985-0

    Book  MATH  Google Scholar 

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Acknowledgements

The research is supported by the Austrian Science Fund (FWF) through the NFN S117-03 project. Implementation was carried out using the open-source C++ library G+smo [19] developed at RICAM.

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Authors and Affiliations

  1. RICAM Linz, Johann Radon Institute, Linz, Austria

    Ulrich Langer & Svetlana Matculevich

  2. St. Petersburg Department of V.A. Steklov Institute of Mathematics RAS, Saint Petersburg, Russia

    Sergey Repin

  3. University of Jyvaskyla, Jyväskylä, Finland

    Sergey Repin

Authors
  1. Ulrich Langer
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  2. Svetlana Matculevich
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  3. Sergey Repin
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Corresponding author

Correspondence to Svetlana Matculevich .

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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., Matculevich, S., Repin, S. (2018). Functional Type Error Control for Stabilised Space-Time IgA Approximations to Parabolic 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_5

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  • DOI: https://doi.org/10.1007/978-3-319-73441-5_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73440-8

  • Online ISBN: 978-3-319-73441-5

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