This talk deals with the time-harmonic wave equation, where the parameter is the wavenumber that sweeps a given interval of interest. We are aim at (i) deriving an approximation to the Helmholtz frequency response map for all the parameter values ranging in the given interval of interest, and at (ii) identifying the possible resonances of the problem.
In principle, those issues could be addressed via the direct numerical evaluation of the frequency response map for the whole range of frequencies. However, this naive approach is prohibitive, because of its unaffordable computational cost.
This talk presents Model order reduction (MOR) methods that aims at alleviating the computational cost, by producing an approximation to the frequency response (the so-called surrogate) that presents the double advantage of being accurate and cheap to evaluate. We look for a rational surrogate: the roots of its denominator will be approximation to the resonances of the problem. The construction relies on the computation of a set of snapshots (offine information) by means of the adaptive finite element method.