  The main focus of the present work is the inclusion of spatial adaptivity for
the snapshot computation in the offline phase of model order reduction
utilizing Proper Orthogonal Decomposition (POD-MOR) for nonlinear parabolic
evolution problems. We consider snapshots which live in different finite
element spaces, which means in a fully discrete setting that the snapshots are
vectors of different length. From a numerical point of view, this leads to the
problem that the usual POD procedure which utilizes a singular value
decomposition of the snapshot matrix, cannot be carried out. In order to
overcome this problem, we here construct the POD model / basis using the
eigensystem of the correlation matrix (snapshot gramian), which is motivated
from a continuous perspective and is set up explicitly e.g. without the
necessity of interpolating snapshots into a common finite element space. It is
an advantage of this approach that the assembling of the matrix only requires
the evaluation of inner products of snapshots in a common Hilbert space. This
allows a great flexibility concerning the spatial discretization of the
snapshots. The analysis for the error between the resulting POD solution and
the true solution reveals that the accuracy of the reduced order solution can
be estimated by the spatial and temporal discretization error as well as the
POD error. Finally, to illustrate the feasibility our approach, we present a
test case of the Cahn-Hilliard system utilizing h-adapted hierarchical meshes
and two settings of a linear heat equation using nested and non-nested grids.
