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

Dinh Bao Phuong Huynh; David J. Knezevic; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 213-251
  • ISSN: 0764-583X

Abstract

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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 eigenfunction “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.

How to cite

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Phuong Huynh, Dinh Bao, Knezevic, David J., and Patera, Anthony T.. "A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 213-251. <http://eudml.org/doc/273211>.

@article{PhuongHuynh2013,
abstract = {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 eigenfunction “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.},
author = {Phuong Huynh, Dinh Bao, Knezevic, David J., Patera, Anthony T.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
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; elliptic equations; component mode systhesis; finite element; order reduction},
language = {eng},
number = {1},
pages = {213-251},
publisher = {EDP-Sciences},
title = {A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation},
url = {http://eudml.org/doc/273211},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Phuong Huynh, Dinh Bao
AU - Knezevic, David J.
AU - Patera, Anthony T.
TI - A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 213
EP - 251
AB - 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 eigenfunction “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.
LA - eng
KW - reduced basis method; reduced basis element method; domain decomposition; Schur complement; elliptic partial differential equations; a posteriori error estimation; component mode synthesis; parametrized systems; elliptic equations; component mode systhesis; finite element; order reduction
UR - http://eudml.org/doc/273211
ER -

References

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