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

Abstract

AbstractWe 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.