Commun. Comput. Phys., 9 (2011), pp. 1-48.


Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries

Gianluigi Rozza 1*

1 Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge MA, 02142-4307, USA.

Received 10 March 2010; Accepted (in revised version) 26 July 2010
Available online 5 August 2010
doi:10.4208/cicp.100310.260710a

Abstract

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) an a posteriori error estimation procedure: rigorous and sharp bounds for the linear-functional outputs of interest and over the potential solution or related quantities of interest like velocity and/or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimum marginal computational cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.

AMS subject classifications: 65Y20, 76B99, 35Q35, 65N15

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Key words: Reduced basis approximation, error bounds, potential flows, Galerkin method, a posteriori error estimation, parametrized geometries.

*Corresponding author.
Email: rozza@mit.edu (G. Rozza)
 

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