An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems

Abstract

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.