  We present a general -- i.e., independent of the underlying equation --
registration procedure for parameterized model order reduction. Given the
spatial domain $\Omega \subset \mathbb{R}^2$ and the manifold $\mathcal{M}= \{
u_{\mu} : \mu \in \mathcal{P} \}$ associated with the parameter domain
$\mathcal{P} \subset \mathbb{R}^P$ and the parametric field $\mu \mapsto
u_{\mu} \in L^2(\Omega)$, our approach takes as input a set of snapshots $\{
u^k \}_{k=1}^{n_{\rm train}} \subset \mathcal{M}$ and returns a
parameter-dependent bijective mapping $\underline{\Phi}: \Omega \times
\mathcal{P} \to \mathbb{R}^2$: the mapping is designed to make the mapped
manifold $\{ u_{\mu} \circ \underline{\Phi}_{\mu}: \, \mu \in \mathcal{P} \}$
more amenable for linear compression methods. In this work, we extend and
further analyze the registration approach proposed in [Taddei, SISC, 2020]. The
contributions of the present work are twofold. First, we extend the approach to
deal with annular domains by introducing a suitable transformation of the
coordinate system. Second, we discuss the extension to general two-dimensional
geometries: towards this end, we introduce a spectral element approximation,
which relies on a partition $\{ \Omega_{q} \}_{q=1} ^{N_{\rm dd}}$ of the
domain $\Omega$ such that $\Omega_1,\ldots,\Omega_{N_{\rm dd}}$ are isomorphic
to the unit square. We further show that our spectral element approximation can
cope with parameterized geometries. We present rigorous mathematical analysis
to justify our proposal; furthermore, we present numerical results for a
heat-transfer problem in an annular domain and for a potential flow past a
rotating airfoil to demonstrate the effectiveness of our method.
