Port Reduction in Component-Based Static Condensation for Parametrized Problems: Approximation and A Posteriori Error Estimation∗

We introduce a port (interface) approximation and a posteriori error bound framework for a general component–based static condensation method in the context of parameter–dependent elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces — the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system — and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation to the true error and bound conditioners — the error in the system approximation or in an output quantity may be bounded with respect to the non-port-reduced full approximation. Our approximation and a posteriori error bound framework is of particular computational relevance for the Static Condensation Reduced Basis Element (SCRBE) method for parameter–dependent partial differential equations. We provide several numerical examples with the SCRBE that serve to both illustrate our port approximation procedure and to demonstrate the efficacy of our port reduction error bounds.