Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs

In this paper we present some heuristic strategies to compute rapid and reliable approximations to stability factors in nonlinear, inf-sup stable parametrized PDEs. The efficient evaluation of these quantities is crucial for the rapid construction of a posteriori error estimates to reduced basis approximations. In this context, stability factors depend on the problem’s solution, and in particular on its reduced basis approximation. Their evaluation becomes therefore very expensive and cannot be performed prior to (and independently of) the construction of the reduced space. As a remedy, we first propose a linearized, heuristic version of the Successive Constraint Method (SCM), providing a suitable estimate – rather than a rigorous lower bound as in the original SCM – of the stability factor. Moreover, for the sake of computational efficiency, we develop an alternative heuristic strategy, which combines a radial basis interpolant, suitable criteria to ensure its positiveness, and an adaptive choice of interpolation points through a greedy procedure. We provide some theoretical results to support the proposed strategies, which are then applied to a set of test cases dealing with parametrized Navier-Stokes equations. Finally, we show that the interpolation strategy is inexpensive to apply and robust even in the proximity of bifurcation points, where the estimate of stability factors is particularly critical.