A Natural–Norm Successive Constraint Method for Inf-Sup Lower Bounds

We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the “linearized” inf-sup statement of the natural–norm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural–norm) provides an economical parameter expansion and local concavity in parameter — a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework — a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural–norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural–norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection–diffusion equation.