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.