True Error Control for the Localized Reduced Basis Method for Parabolic Problems

Chapter
Part of the MS&A book series (MS&A, volume 17)

Abstract

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques (Makridakis and Nochetto, SIAM J. Numer. Anal. 41(4):1585–1594, 2003. doi:10.1137/S0036142902406314; Lakkis and Makridakis, Math. Comput. 75(256):1627–1658, 2006. doi:10.1090/S0025-5718-06-01858-8; Demlow et al., SIAM J. Numer. Anal. 47(3):2157–2176, 2009. doi:10.1137/070708792; Georgoulis et al., SIAM J. Numer. Anal. 49(2):427–458, 2011. doi:10.1137/080722461). In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method (Ohlberger and Schindler, SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015. doi:10.1137/151003660). Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Mario Ohlberger
    • 1
  • Stephan Rave
    • 1
  • Felix Schindler
    • 1
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany

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