A Two-Step Certified Reduced Basis Method

In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element "truth" discretization of dimension ${\mathcal{N}} $ an intermediate RB model of dimension $N\ll {\mathcal{N}}$ . In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M?N. The construction of the DRB model is effected at cost ${\mathcal{O}}(N)$ and in particular at cost independent of ${\mathcal{N}}$ ; subsequent evaluation of the DRB model may then be effected at cost ${\mathcal{O}}(M)$ . The DRB model comprises both the DRB output and a rigorous a posteriori error bound for the error in the DRB output with respect to the truth discretization. The new approach is of particular interest in two contexts: focus calculations and hp-RB approximations. In the former the new approach serves to reduce online cost, M?N: the DRB model is restricted to a slice or subregion of a larger parameter domain associated with the intermediate RB model. In the latter the new approach enlarges the class of problems amenable to hp-RB treatment by a significant reduction in offline (precomputation) cost: in the development of the hp parameter domain partition and associated "local" (now derived) RB models the finite element truth is replaced by the intermediate RB model. We present numerical results to illustrate the new approach.