  In this paper, we construct a class of Mixed Generalized Multiscale Finite
Element Methods for the approximation on a coarse grid for an elliptic problem
in thin two-dimensional domains. We consider the elliptic equation with
homogeneous boundary conditions on the domain walls. For reference solution of
the problem, we use a Mixed Finite Element Method on a fine grid that resolves
complex geometry on the grid level. To construct a lower dimensional model, we
use the Mixed Generalized Multiscale Finite Element Method, which is based on
some multiscale basis functions for velocity fields. The construction of the
basis functions is based on the local snapshot space that takes all possible
flows on the interface between coarse cells into account. In order to reduce
the size of the snapshot space and obtain the multiscale approximation, we
solve a local spectral problem to identify dominant modes in the snapshot
space. We present a convergence analysis of the presented multiscale method.
Numerical results are presented for two-dimensional problems in three testing
geometries along with the errors associated to different numbers of the
multiscale basis functions used for the velocity field. Numerical
investigations are conducted for problems with homogeneous and heterogeneous
properties respectively.
