On the stability of the reduced basis method for Stokes equations in parametrized domains

Gianluigi Rozza, Karen Veroy

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140 Citations (Scopus)

Abstract

We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global reduced basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) the off-line/on-line computational procedures decoupling the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate an output of interest - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Particular attention is given (i) to the pressure treatment of incompressible Stokes problem; (ii) to find an equivalent inf-sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem; (iii) to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions; (iv) to reduce computational costs in order to allow real-time solution of parametrized problem.

Original languageEnglish
Pages (from-to)1244-1260
Number of pages17
JournalComputer Methods in Applied Mechanics and Engineering
Volume196
Issue number7
DOIs
Publication statusPublished - 10 Jan 2007
Externally publishedYes

Bibliographical note

Funding Information:
We acknowledge Prof. A.T. Patera of MIT, Prof. A. Quarteroni of EPF-Lausanne and MOX-Politecnico di Milano and Prof. Y. Maday of University Paris VI, Dr. D. Rovas (University of Illinois), Dr. C. Prud’homme (EPFL) and Dr. N.C. Nguyen (National University of Singapore) for suggestions, very helpful discussion, comments and insights, Prof. F. Saleri (MOX-Politecnico di Milano) for providing the original finite element Stokes solver. G. Rozza acknowledges the support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2002-00270 HaeMOdel. This work was supported also by Swiss National Science Foundation (PBEL2-111646), by DARPA and ASFOR under grant F4920-03-1-0356, by DARPA and GEAE under grant F49620-03-1-0439 and by Singapore–MIT Alliance (SMA).

Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

Keywords

  • Algebraic stability
  • Approximation stability
  • Galerkin approximation
  • Gram-Schmidt basis orthogonalization
  • Inf-sup condition
  • Parametrized Stokes equations
  • Reduced basis methods
  • Supremizer

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