Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds
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"Certified real‐time solution of the..." refers background in this paper
...Our key new ingredients are appropriate approximations and associated o ine–online computational procedures for calculation of (a) the dual norm of the requisite residuals, (b) an upper bound for the ‘L4( ) − H 1( )’ Sobolev embedding continuity constant [15, 16], (c) a lower bound for the Babu ska inf–sup stability factor, and (d) the adjoint contributions associated with the output....
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...As our point of departure, we note [15, 16] that =(2=̂min), where (̂; ̂)∈ (R+; Ỹ ) satis es the Euler–Lagrange equation (̂; v)Y = ̂ ∫ ̂ĵĵivi, ∀v∈ Ỹ , ‖̂‖L4( ) = 1, and (̂min; ̂min) denotes the ground state....
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...is a Sobolev embedding constant [15, 16] and ‖v‖Lp( ) ≡ ( ∫ (vivi) p=2)1=p....
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"Certified real‐time solution of the..." refers methods in this paper
...We next introduce the key parameters required by the BRR theory [11–13]....
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...To construct our a posteriori estimators, we invoke the Brezzi–Rappaz–Raviart (BRR) theory for analysis of variational approximations of nonlinear partial di erential equations [11–14]....
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"Certified real‐time solution of the..." refers background or methods in this paper
...§We choose Y to be the (discretely) incompressible space of dimension N=4762 derived from a Taylor–Hood P2 − P1 approximation space [8] with 5538 velocity and 776 pressure degrees-of-freedom....
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...) RB approximation [2–7] takes advantage of the dimension reduction a orded by the (smooth) parametrically-induced solution manifold: successful application to the incompressible Navier–Stokes equations [8–10] is well documented; our emphasis is thus on the development and application of rigorous a posteriori error estimation procedures....
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