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We introduce several spatially adaptive model order reduction approachestailored to non-coercive elliptic boundary value problems, specifically,parametric-in-frequency Helmholtz problems. The offline information is computedby means of adaptive finite elements, so that each snapshot lives in adifferent discrete space that resolves the local singularities of theanalytical solution and is adjusted to the considered frequency value. Arational surrogate is then assembled adopting either a least-squares or aninterpolatory approach, yielding function-valued version of the standardrational interpolation method ($\mathcal{V}$-SRI) and the minimal rationalinterpolation method (MRI). In the context of building an approximation forlinear or quadratic functionals of the Helmholtz solution, we perform severalnumerical experiments to compare the proposed methodologies. Our simulationsshow that, for interior resonant problems (whose singularities are encoded bypoles on the real axis), the spatially adaptive $\mathcal{V}$-SRI and MRI workcomparably well. Instead, when dealing with exterior scattering problems, whosefrequency response is mostly smooth, the $\mathcal{V}$-SRI method seems to bethe best-performing one.

Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots