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The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case

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Spectral and High Order Methods for Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 76))

Abstract

This work focuses on the reduced basis element method applied to the steady Stokes problem with geometric parameter dependence [2, 4]. We present a decoupling of the operators involved in the steady Stokes problem, which together with empirical interpolation [1] allows for complete separation of the offline-online complexity for the nonaffine case. We present numerical results from a hierarchical flow system in two dimensions, where both pipes and bifurcations are used as building blocks.

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References

  1. M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera. An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Serie I, 339, 667–672 (2004)

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  2. A. E. Løvgren, Y. Maday, and E. M. Rønquist. A reduced basis element method for the steady Stokes problem. M2AN, 40(3), 529–552 (2006)

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  3. Y. Maday and E. M. Rønquist. The reduced-basis element method: Application to a thermal fin problem. SIAM J. Sci. Comput., 26(1), 240–258 (2004)

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  4. G. Rozza and A. Quarteroni. Numerical solution of parametrized Navier–Stokes equations by reduced basis methods. Numer. Meth. for PDEs, 23(4), 923–948 (2007)

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Acknowledgements

This work has been supported by the Research Council of Norway through a Centre of Excellence grant to the Center for Biomedical Computing.

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Authors and Affiliations

  1. Simula Research Laboratory, Center for Biomedical Computing, 134, 1325, Lysaker, Norway

    A. E. Løvgren

  2. Laboratoire J.-L. Lions, UMR 7598, Université Pierre et Marie Curie-Paris6, Paris, 75005, France

    Y. Maday

  3. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Y. Maday

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    E. M. Rønquist

Authors
  1. A. E. Løvgren
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  2. Y. Maday
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  3. E. M. Rønquist
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Corresponding author

Correspondence to A. E. Løvgren .

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Editors and Affiliations

  1. , Division of Applied Mathematics, Brown University, George Street 182, Providence, 02912, Rhode Island, USA

    Jan S. Hesthaven

  2. , Department of Mathematical Sciences, Norwegian University of Science and Tech, Trondheim, 7491, Norway

    Einar M. Rønquist

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Løvgren, A.E., Maday, Y., Rønquist, E.M. (2011). The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_22

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