  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 ${\Phi}: \Omega \times \mathcal{P} \to
\mathbb{R}^2$: the mapping is designed to make the mapped manifold $\{ u_{\mu}
\circ {\Phi}_{\mu}: \, \mu \in \mathcal{P} \}$ 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, a potential flow past a rotating symmetric
airfoil, and an inviscid transonic compressible flow past a non-symmetric
airfoil, to demonstrate the effectiveness of our method.
