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 anely parametrized partial dierential 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 ecacy 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{diusi on equation.

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