  We consider model order reduction by proper orthogonal decomposition (POD)
for parametrized partial differential equations, where the underlying snapshots
are computed with adaptive finite elements. We address computational and
theoretical issues arising from the fact that the snapshots are members of
different finite element spaces. We propose a method to create a POD-Galerkin
model without interpolating the snapshots onto their common finite element
mesh. The error of the reduced-order solution is not necessarily Galerkin
orthogonal to the reduced space created from space-adapted snapshot. We analyze
how this influences the error assessment for POD-Galerkin models of linear
elliptic boundary value problems. As a numerical example we consider a
two-dimensional convection-diffusion equation with a parametrized convective
direction. To illustrate the applicability of our techniques to non-linear
time-dependent problems, we present a test case of a two-dimensional viscous
Burgers equation with parametrized initial data.
