Comptes Rendus
Partial differential equations/Numerical analysis
An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems
Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1188-1194.

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.

Nous proposons une nouvelle strategie pour stabiliser l'approximation d'un problème de diffusion–transport avec transport dominant par une méthode de bases réduites. Cette strategie, opérée en ligne, est indépendante de la technique « haute fidélité » utilisée « hors ligne » ; elle trouve son inspiration dans la méthode de la viscosité spectrale évanescente. Par une diagonalisation sur l'espace de base réduite, on introduit une nouvelle base modale, qui permet d'ajouter au problème réduit un terme de viscosité évanescent sur les modes suffisant pour stabiliser l'approximation. Une méthode de rectification de la solution (semblable aux techniques de filtrage spectral) de ce problème est enfin opérée afin d'améliorer la précision de cette approximation. Les résultats numériques obtenus pour un problème avec transport dominant dont l'intensité est parametrisée montrent que l'approximation réduite résultante est stable et précise sur tout l'intervalle des paramètres.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.008
Yvon Maday 1, 2, 3; Andrea Manzoni 4; Alfio Quarteroni 4

1 Sorbonne Universités, UPMC Université Paris-6 and CNRS UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
2 Institut universitaire de France, France
3 Division of Applied Mathematics, Brown University, Providence RI, USA
4 CMCS–MATHICSE–SB, École polytechnique fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
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     title = {An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems},
     journal = {Comptes Rendus. Math\'ematique},
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Yvon Maday; Andrea Manzoni; Alfio Quarteroni. An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1188-1194. doi : 10.1016/j.crma.2016.10.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.008/

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