Maday, Yvon; Manzoni, Andrea; Quarteroni, Alfio An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems. (Une stratégie intrinsèque de stabilisation en ligne pour l’approximation bases réduites de problèmes paramètrés avec transport dominant.) (English. Abridged French version) Zbl 1353.65093 C. R., Math., Acad. Sci. Paris 354, No. 12, 1188-1194 (2016). Summary: We propose a new, black-box online stabilization strategy for reduced basis (RB) approximations of parameter-dependent advection-diffusion problems in the advection-dominated case. Our goal is to stabilize the RB problem irrespectively of the stabilization (if any) operated on the high-fidelity (e.g., finite element) approximation, provided a set of stable RB functions have been computed. Inspired by the spectral vanishing viscosity method, our approach relies on the transformation of the basis functions into modal basis, then on the addition of a vanishing viscosity term over the high RB modes, and on a rectification stage – prompted by the spectral filtering technique – to further enhance the accuracy of the RB approximation. Numerical results dealing with an advection-dominated problem parametrized with respect to the diffusion coefficient show the accuracy of the RB solution on the whole parametric range. Cited in 7 Documents MSC: 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L02 First-order hyperbolic equations 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs Keywords:stabilization; reduced basis approximations; advection-diffusion problem; finite element; spectral vanishing viscosity method; spectral filtering technique; numerical results Software:redbKIT PDFBibTeX XMLCite \textit{Y. Maday} et al., C. R., Math., Acad. Sci. Paris 354, No. 12, 1188--1194 (2016; Zbl 1353.65093) Full Text: DOI References: [1] Chakir, R.; Maday, Y., A two-grid finite-element/reduced basis scheme for the approximation of the solution of parametric dependent PDE, C. R. Acad. Sci. Paris, Ser. I, 347, 435-440 (2009) · Zbl 1161.65083 [2] Ern, A.; Guermond, J.-L., Theory and Practice of Finite Elements (2004), Springer-Verlag: Springer-Verlag New York [3] Ern, A.; Stephansen, A., A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods, J. Comput. Math., 26, 488-510 (2008) · Zbl 1174.65034 [4] Gottlieb, D.; Shu, C. W.; Solomonoff, A.; Vandeven, H., On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math., 43, 1-2, 81-98 (1992) · Zbl 0781.42022 [5] Hesthaven, J. S.; Rozza, G.; Stamm, B., Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics (2016), Springer · Zbl 1329.65203 [6] Herrero, H.; Maday, Y.; Pla, F., RB (Reduced basis) for RB (Rayleigh-Bénard), Comput. Methods Appl. Mech. Eng., 261-262, 132-141 (2013) · Zbl 1286.76084 [7] Hughes, T. J.R.; Feijóo, G.; Mazzei, L.; Quincy, J.-B., The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166, 3-24 (1998) · Zbl 1017.65525 [8] Maday, Y.; Tadmor, E., Analysis of the spectral vanishing viscosity method for periodic conservation laws, SIAM J. Numer. Anal., 26, 4, 854-870 (1989) · Zbl 0678.65066 [9] Quarteroni, A., Numerical Models for Differential Problems, Modeling, Simulation and Applications (MS&A), vol. 8 (2014), Springer-Verlag: Springer-Verlag Italy [10] Quarteroni, A.; Manzoni, A.; Negri, F., Reduced Basis Methods for Partial Differential Equations. An Introduction, Unitext Series, vol. 92 (2016), Springer · Zbl 1337.65113 [11] Rozza, G.; Pacciarini, P., Stabilized reduced basis method for parametrized advection-diffusion PDEs, Comput. Methods Appl. Mech. Eng., 274, 1, 1-18 (2014) · Zbl 1296.65165 [12] Verfürth, R., Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal., 43, 4, 1766-1782 (2005) · Zbl 1099.65100 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.