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A two-step certified reduced basis method. (English) Zbl 1244.65172

Summary: 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 \(M\leq N\). 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, \(M\ll N\): 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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Software:

libMesh
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References:

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