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A Static condensation Reduced Basis Element method :approximation and a posteriori error estimation

Published online by Cambridge University Press:  23 November 2012

Dinh Bao Phuong Huynh ,
David J. Knezevic  and
Anthony T. Patera
Dinh Bao Phuong Huynh
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu
David J. Knezevic
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu School of Engineering and Applied Sciences, Harvard University, Cambridge, 02138 MA, USA; dknezevic@seas.harvard.edu; patera@mit.edu
Anthony T. Patera
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu

Abstract

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We propose a new reduced basis element-cum-component mode synthesis approach forparametrized elliptic coercive partial differential equations. In the Offline stage weconstruct a Library of interoperable parametrized reference componentsrelevant to some family of problems; in the Online stage we instantiate andconnect reference components (at ports) to rapidly form and query parametricsystems. The method is based on static condensation at the interdomainlevel, a conforming eigenfunction “port” representation at the interface level, andfinally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at theintradomain level. We show under suitable hypotheses that the RB Schur complement is closeto the FE Schur complement: we can thus demonstrate the stability of the discreteequations; furthermore, we can develop inexpensive and rigorous (system-level) aposteriori error bounds. We present numerical results for model many-parameterheat transfer and elasticity problems with particular emphasis on the Online stage; wediscuss flexibility, accuracy, computational performance, and also the effectivity of thea posteriori error bounds.

Keywords

Reduced Basis methodReduced Basis Element methoddomain decompositionSchur complementelliptic partial differential equationsa posteriori error estimationcomponent mode synthesisparametrized systems

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 213 - 251
Copyright
© EDP Sciences, SMAI, 2012

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