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Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. (English) Zbl 1134.76326
Summary: 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 ‘\(L^{4}(\Omega)-H^{1}(\Omega)\)’ 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.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M30 Variational methods applied to problems in fluid mechanics
65N15 Error bounds for boundary value problems involving PDEs
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