Research Article

Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds

  1. K. Veroy,
  2. A. T. Patera*

Article first published online: 31 JAN 2005

DOI: 10.1002/fld.867

Issue

International Journal for Numerical Methods in Fluids

International Journal for Numerical Methods in Fluids

Special Issue: 8th ICFD Conference on Numerical Methods for Fluid Dynamics

Volume 47, Issue 8-9, pages 773–788, 20 - 30 March 2005

Additional Information

How to Cite

Veroy, K. and Patera, A. T. (2005), Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds. International Journal for Numerical Methods in Fluids, 47: 773–788. doi: 10.1002/fld.867

Author Information

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 3-266, Cambridge, MA 02139, U.S.A.

*Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 3-266, Cambridge, MA 02139, U.S.A.

Publication History

  1. Issue published online: 4 MAR 2005
  2. Article first published online: 31 JAN 2005
  3. Manuscript Accepted: 11 NOV 2004
  4. Manuscript Revised: 8 NOV 2004
  5. Manuscript Received: 27 APR 2004

Funded by

  • DARPA/AFOSR. Grant Number: F49620-03-1-0356
  • DARPA/GEAE. Grant Number: F49620-03-1-0439
  • Singapore-MIT Alliance

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Keywords:

  • reduced-basis;
  • a posteriori error estimation;
  • output bounds;
  • offline–online procedures;
  • incompressible Navier–Stokes;
  • natural convection;
  • parametrized partial differential equations

Abstract

We present a technique for the evaluation of linear-functional outputs of parametrized elliptic partial differential equations in the context of deployed (in service) systems. Deployed systems require real-time and certified output prediction in support of immediate and safe (feasible) action. The two essential components of our approach are (i) rapidly, uniformly convergent reduced-basis approximations, and (ii) associated rigorous and sharp a posteriori error bounds; in both components we exploit affine parametric structure and offline–online computational decompositions to provide real-time deployed response. In this paper we extend our methodology to the parametrized steady incompressible Navier–Stokes equations.

We invoke the Brezzi–Rappaz–Raviart theory for analysis of variational approximations of non-linear partial differential equations to construct rigorous, quantitative, sharp, inexpensive a posteriori error estimators. The crucial new contribution is offline–online computational procedures for calculation of (a) the dual norm of the requisite residuals, (b) an upper bound for the ‘L4(Ω)-H1(Ω)’ Sobolev embedding continuity constant, (c) a lower bound for the Babuška inf–sup stability ‘constant,’ and (d) the adjoint contributions associated with the output. Numerical results for natural convection in a cavity confirm the rapid convergence of the reduced-basis approximation, the good effectivity of the associated a posteriori error bounds in the energy and output norms, and the rapid deployed response. Copyright © 2005 John Wiley & Sons, Ltd.

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