A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) no. 1, pp. 213-251.

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.

DOI : https://doi.org/10.1051/m2an/2012022
Classification : 35J25,  65N30,  65D99
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
     author = {Phuong Huynh, Dinh Bao and Knezevic, David J. and Patera, Anthony T.},
     title = {A {Static} condensation {Reduced} {Basis} {Element} method : approximation and a posteriori error estimation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {213--251},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     doi = {10.1051/m2an/2012022},
     zbl = {1276.65082},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012022/}
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
DA  - 2013///
SP  - 213
EP  - 251
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012022/
UR  - https://zbmath.org/?q=an%3A1276.65082
UR  - https://doi.org/10.1051/m2an/2012022
DO  - 10.1051/m2an/2012022
LA  - en
ID  - M2AN_2013__47_1_213_0
ER  - 
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 - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) no. 1, pp. 213-251. doi : 10.1051/m2an/2012022. http://www.numdam.org/articles/10.1051/m2an/2012022/

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