Journal
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
Volume 47, Issue 1, Pages 213-251Publisher
EDP SCIENCES S A
DOI: 10.1051/m2an/2012022
Keywords
Reduced Basis method; Reduced Basis Element method; domain decomposition; Schur complement; elliptic partial differential equations; a posteriori error estimation; component mode synthesis; parametrized systems
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Funding
- OSD/AFOSR/MURI [FA9550-09-1-0613]
- ONR [N00014-11-0713]
- MIT-Singapore International Design Center
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We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigen-function port representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.
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