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An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows

Published online by Cambridge University Press:  15 July 2014

Andrea Manzoni
Andrea Manzoni*
Affiliation:
SISSA Mathlab – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy. amanzoni@sissa.it
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Abstract

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.

Keywords

Reduced Basis Methodparametrized Navier–Stokes equationssteady incompressible fluidsa posteriori error estimationapproximation stability
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1199 - 1226
Copyright
© EDP Sciences, SMAI, 2014

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