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.
Mots-clés : reduced basis method, reduced basis element method, domain decomposition, Schur complement, elliptic partial differential equations, a posteriori error estimation, component mode synthesis, parametrized systems
@article{M2AN_2013__47_1_213_0,
author = {Phuong Huynh, Dinh Bao and Knezevic, David J. and Patera, Anthony T.},
title = {A {Static} condensation {Reduced} {Basis} {Element} method : approximation and \protect\emph{a posteriori }error estimation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {213--251},
publisher = {EDP-Sciences},
volume = {47},
number = {1},
year = {2013},
doi = {10.1051/m2an/2012022},
zbl = {1276.65082},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012022/}
}
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 PY - 2013 SP - 213 EP - 251 VL - 47 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012022/ DO - 10.1051/m2an/2012022 LA - en ID - M2AN_2013__47_1_213_0 ER -
%0 Journal Article %A Phuong Huynh, Dinh Bao %A Knezevic, David J. %A Patera, Anthony T. %T A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 213-251 %V 47 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012022/ %R 10.1051/m2an/2012022 %G en %F M2AN_2013__47_1_213_0
Phuong Huynh, Dinh Bao; Knezevic, David J.; Patera, Anthony T. A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 213-251. doi : 10.1051/m2an/2012022. http://archive.numdam.org/articles/10.1051/m2an/2012022/
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