  In this paper, we consider flow and transport problems in thin domains. The
mathematical model considered in the paper is described by a system of
equations for velocity, pressure, and concentration, where the flow is
described by the Stokes equations and the transport is described by an unsteady
convection-diffusion equation with non-homogeneous boundary conditions on walls
(reactive boundaries). We start with the finite element approximation of the
problem on unstructured grids and use it as a reference solution for two and
three-dimensional model problems. Fine grid approximation resolves complex
geometries on the grid level and leads to a large discrete system of equations
that is computationally expensive to solve. To reduce the size of the discrete
systems, we develop a multiscale model reduction technique, where we construct
local multiscale basis functions to generate a lower-dimensional model on a
coarse grid. The proposed multiscale model reduction is based on the
Discontinuous Galerkin Generalized Multiscale Finite Element Method
(DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the
snapshot space for each interface between coarse grid cells to capture possible
flows. For the reduction of the snapshot space size, we perform a dimension
reduction via a solution of the local spectral problem and use eigenvectors
corresponding to the smallest eigenvalues as multiscale basis functions for the
approximation on the coarse grid. For the transport problem, we construct
multiscale basis functions for each interface between coarse grid cells and
present additional basis functions to capture non-homogeneous boundary
conditions on walls. Finally, we will present some numerical simulations for
three test geometries for two and three-dimensional problems to demonstrate the
method's performance.
