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A natural-norm Successive Constraint Method for inf-sup lower bounds

Author(s)
Huynh, Dinh Bao Phuong; Knezevic, David; Chen, Y.; Hesthaven, J. S.; Patera, Anthony T.
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Abstract
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.
Date issued
2010-06
URI
http://hdl.handle.net/1721.1/64476
Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Journal
Computer Methods in Applied Mechanics and Engineering
Publisher
Elsevier ScienceDirect
Citation
Huynh, D.B.P. et al. “A natural-norm Successive Constraint Method for inf-sup lower bounds.” Computer Methods in Applied Mechanics and Engineering 199.29-32 (2010) : 1963-1975.Copyright © 2010, Elsevier
Version: Author's final manuscript
ISSN
0045-7825

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