A new certification framework for the port-reduced static condensation reduced basis element method

In this talk we introduce a new certification framework for the port-reduced static condensation reduced basis element (PR-SCRBE) method [1, 3], which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides. In an offline computational stage we construct a library of interoperable parametrized reference components; in the subsequent online stage we instantiate and connect the components at the interfaces/ports to form a system of components. To compute a "truth" finite element approximation of the (say) coercive elliptic partial differential equation on the component based system we use a domain decomposition approach. For an efficient simulation we employ two different types of model reduction — a reduced basis (RB) approximation within the interior of the component [3] and empirical port reduction on the ports where the components connect [1]. We demonstrate the well-posedness of the PR-SCRBE approximation and introduce a new certification framework. To assess the quality of the port reduction we use conservative fluxes [2]. Next we adapt the standard estimators from RB methods to the SCRBE setting to derive an a posteriori error bound for the RB-error contribution. In order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for PR-SCRBE we use techniques from multi-scale methods. Finally, we prove that the effectivity of the derived estimator can be bounded. We provide several numerical experiments to show that the derived a posteriori error estimator provides a sharp and effective bound of the H-error between the PR-SCRBE approximation and the "truth" finite element solution. Moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational (online) costs.