{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T21:50:25Z","timestamp":1791409825129,"version":"4.3.3"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms\u2014local parameter-dependent norms\u2014to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi\u2013Rappaz\u2013Raviart existence and uniqueness theorem in the presence of two distinct norms. This allows us to relax the existence condition and to sharpen the field variable error bound. We also provide a robust algorithm to compute the Sobolev embedding constants involved in the error bound and in the inf-sup lower bound computation. We apply our method to a steady natural convection problem in a closed cavity, with a Grashof number varying from 10 to $10^7$.<\/jats:p>","DOI":"10.1137\/060674181","type":"journal-article","created":{"date-parts":[[2008,4,23]],"date-time":"2008-04-23T18:23:16Z","timestamp":1208974996000},"page":"2039-2067","source":"Crossref","is-referenced-by-count":55,"title":["Reduced Basis Error Bound Computation of Parameter-Dependent Navier\u2013Stokes Equations by the Natural Norm Approach"],"prefix":"10.1137","volume":"46","author":[{"given":"Simone","family":"Deparis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,23]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.2514\/3.7539"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1006\/mssp.1996.0027"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492901000010"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01395985"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"H. 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Ravindran,\n                      A reduced basis method for control problems governed by PDEs\n                      , in Control and Estimation of Distributed Parameter Systems, W. Desch, F. Kappel, and K. Kunisch, eds., Birkh\u00e4user, Cambridge, MA, 1998, pp. 153\u2013168.","DOI":"10.1007\/978-3-0348-8849-3_12"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.5943"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500366124"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2006021"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"A. E. L\u00f8vgren, Y. Maday, and E. M. R\u00f8nquist,\n                      The reduced basis element method for fluid flows\n                      , in Analysis and Simulation of Fluid Dynamics, C. Calgaro, J.F. Coulombel, and T. Goudon, eds., Adv. Math. Fluid Mech. 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Nguyen,\n                      Reduced-Basis Approximation and A Posteriori Error Bounds for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis\n                      , Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore, 2005."},{"key":"R17","unstructured":"N. C. Nguyen, K. Veroy, and A. T. Patera,\n                      Certified real-time solution of parametrized partial differential equations\n                      , in Handbook of Materials Modeling, S. Yip, ed., Springer, New York, 2005, pp. 1523\u20131558."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.2514\/3.50778"},{"key":"R19","unstructured":"A. T. 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