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

Ohlberger M, Rave S, Schindler F

Forschungsartikel (Buchbeitrag) | Peer reviewed

Zusammenfassung

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.

Details zur Publikation

Herausgeber*innenBenner P., Ohlberger M., Patera A., Rozza G., Urban K.
BuchtitelModel Reduction of Parametrized Systems (Band 2016)
Seitenbereich169-182
VerlagSpringer International Publishing
ErscheinungsortCham
Titel der ReiheMS&A (Modeling, Simulation and Applications)
Nr. in Reihe17
StatusVeröffentlicht
Veröffentlichungsjahr2017
Sprache, in der die Publikation verfasst istEnglisch
ISBN978-3-319-58785-1
DOI10.1007/978-3-319-58786-8_11
Link zum Volltexthttps://arxiv.org/abs/1606.09216

Autor*innen der Universität Münster

Ohlberger, Mario
Professur für Angewandte Mathematik, insbesondere Numerik (Prof. Ohlberger)
Center for Nonlinear Science (CeNoS)
Center for Multiscale Theory and Computation (CMTC)
Rave, Stephan
Mathematisches Institut
Schindler, Felix Tobias
Professur für Angewandte Mathematik, insbesondere Numerik (Prof. Ohlberger)