We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier-Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039-2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923-948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773-788]) to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi-Rappaz-Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.
Mots clés : reduced basis method, parametrized Navier-Stokes equations, steady incompressible fluids, a posteriori error estimation, approximation stability
@article{M2AN_2014__48_4_1199_0,
author = {Manzoni, Andrea},
title = {An efficient computational framework for reduced basis approximation and \protect\emph{a posteriori }error estimation of parametrized {Navier-Stokes} flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1199--1226},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
doi = {10.1051/m2an/2014013},
zbl = {1301.76025},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014013/}
}
TY - JOUR AU - Manzoni, Andrea TI - An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1199 EP - 1226 VL - 48 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014013/ DO - 10.1051/m2an/2014013 LA - en ID - M2AN_2014__48_4_1199_0 ER -
%0 Journal Article %A Manzoni, Andrea %T An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1199-1226 %V 48 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014013/ %R 10.1051/m2an/2014013 %G en %F M2AN_2014__48_4_1199_0
Manzoni, Andrea. An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 1199-1226. doi : 10.1051/m2an/2014013. http://www.numdam.org/articles/10.1051/m2an/2014013/
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