# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] , , . Certified real-time solution of parametrized partial differential equations. In Handbook of Materials Modeling. Kluwer Academic Publishing: Dordrecht, 2005, to appear. [2] Almroth, AIAA Journal 16 pp 525– (1978) [3] Fink, Zeitschrift für Angewandte Mathematic und Mechanik 63 pp 21– (1983) [4] Machiels, Comptes Rendus de l’Académie des Sciences Paris, Série I 331 pp 153– (2000) [5] Maday, Comptes Rendus de l’Académie des Sciences Paris, Série I 335 pp 289– (2002) · Zbl 1009.65066 · doi:10.1016/S1631-073X(02)02466-4 [6] Noor, AIAA Journal 18 pp 455– (1980) [7] Porsching, Mathematics of Computation 45 pp 487– (1985) [8] . Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press: Boston, 1989. [9] Ito, Journal of Computational Physics 143 pp 403– (1988) [10] Peterson, SIAM Journal on Scientific and Statistical Computing 10 pp 777– (1989) [11] Brezzi, Numerische Mathematik 36 pp 1– (1980) [12] , . Numerical analysis for nonlinear and bifurcation problems. In Handbook of Numerical Analysis, Vol. V, , (eds), Techniques of Scientific Computing (Part 2). Elsevier Science B.V.: Amsterdam, 1997; 487-637. [13] , . Finite Element Approximation of the Navier-Stokes Equations. Springer: Berlin, 1986. · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 [14] , . A reduced basis method for control problems governed by PDEs. In Control and Estimation of Distributed Parameter Systems, , , (eds). Birkhäuser: Boston, 1998; 153-168. · Zbl 0908.93025 · doi:10.1007/978-3-0348-8849-3_12 [15] Talenti, Annali di Matematica Pura ed Applicata 110 pp 353– (1976) [16] Trudinger, Journal of Mathematics and Mechanics 17 pp 473– (1967) [17] Veroy, Comptes Rendus de l’Académie des Sciences Paris, Série I 337 pp 619– (2003) · Zbl 1036.65075 · doi:10.1016/j.crma.2003.09.023 [18] Prud’homme, Journal of Fluids Engineering 124 pp 70– (2002) [19] , , , . A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2003-3847, June 2003. [20] Gelfgat, Journal of Fluid Mechanics 388 pp 315– (1999) [21] Skeldon, Journal of Crystal Growth 162 pp 95– (1996) [22] Machiels, Journal of Computational Physics 172 pp 401– (2001) · Zbl 0984.65100 [23] Balmes, Mechanical Systems and Signal Processing 10 pp 381– (1996) [24] Giles, SIAM Review 42 pp 247– (2000) [25] , , , . Adaptive strategies and error control for computing material forces in fracture mechanics. Technical Report, Chalmers Finite Element Center, Chalmers University of Technology, Göteborg, Sweden, December 2002.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.