A new certification framework for the port reduced static condensation reduced basis element method. (English) Zbl 1425.65179
Summary: In this paper we introduce a new certification framework for the port-reduced static condensation reduced basis element (PR-SCRBE) method, 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 [D. B. P. Huynh et al., ESAIM, Math. Model. Numer. Anal. 47, No. 1, 213–251 (2013; Zbl 1276.65082); Comput. Methods Appl. Mech. Eng. 259, 197–216 (2013; Zbl 1286.65160)] and empirical port reduction [J. L. Eftang and A. T. Patera, Int. J. Numer. Methods Eng. 96, No. 5, 269–302 (2013; Zbl 1352.65495)] on the ports where the components connect. 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. We adapt the standard estimators from RB methods to the SCRBE setting to derive an a posteriori error estimator 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 adapt techniques from multi-scale methods and component mode synthesis. Finally, we prove that the effectivity of the derived estimator can be bounded. We provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator. Moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational (online) costs.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
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