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

Authors

  • K. Veroy,

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

    Corresponding author
    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.
    Search for more papers by this author

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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