A certified natural-norm successive constraint method for parametric inf–sup lower bounds

Abstract

We present a certified version of the Natural-Norm Successive Constraint Method (cNNSCM) for fast and accurate Inf–Sup lower bound evaluation of parametric operators. Successive Constraint Methods (SCM) are essential tools for the construction of a lower bound for the inf–sup stability constants which are required in a posteriori error analysis of reduced basis approximations. They utilize a Linear Program (LP) relaxation scheme incorporating continuity and stability constraints. The natural-norm approach linearizes a lower bound of the inf–sup constant as a function of the parameter. The Natural-Norm Successive Constraint Method (NNSCM) combines these two aspects. It uses a greedy algorithm to select SCM control points which adaptively construct an optimal decomposition of the parameter domain, and then apply the SCM on each domain.Unfortunately, the NNSCM produces no guarantee for the quality of the lower bound. Through multiple rounds of optimal decomposition, the new cNNSCM provides an upper bound in addition to the lower bound and lets the user control the gap, thus the quality of the lower bound. The efficacy and accuracy of the new method is validated by numerical experiments.