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.

Classification: 35J25, 65N30, 65D99

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

@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 a posteriori error estimation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {47}, number = {1}, year = {2013}, pages = {213-251}, doi = {10.1051/m2an/2012022}, zbl = {1276.65082}, language = {en}, url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 213-251. doi : 10.1051/m2an/2012022. http://www.numdam.org/item/M2AN_2013__47_1_213_0/

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