  In this paper, we consider local multiscale model reduction for problems with
multiple scales in space and time. We developed our approaches within the
framework of the Generalized Multiscale Finite Element Method (GMsFEM) using
space-time coarse cells. The main idea of GMsFEM is to construct a local
snapshot space and a local spectral decomposition in the snapshot space.
Previous research in developing multiscale spaces within GMsFEM focused on
constructing multiscale spaces and relevant ingredients in space only. In this
paper, our main objective is to develop a multiscale model reduction framework
within GMsFEM that uses space-time coarse cells. We construct space-time
snapshot and offline spaces. We compute these snapshot solutions by solving
local problems. A complete snapshot space will use all possible boundary
conditions; however, this can be very expensive. We propose using randomized
boundary conditions and oversampling. We construct the local spectral
decomposition based on our analysis, as presented in the paper. We present
numerical results to confirm our theoretical findings and to show that using
our proposed approaches, we can obtain an accurate solution with low
dimensional coarse spaces. We remark that the proposed method is a significant
extension compared to existing methods, which use coarse cells in space only
because of (1) the parabolic nature of cell solutions, (2) extra degrees of
freedom associated with space-time cells, and (3) local boundary conditions in
space-time cells.
