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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

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.

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

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