  In this paper, we present a new multiscale model reduction technique for the
Stokes flows in heterogeneous perforated domains. The challenge in the
numerical simulations of this problem lies in the fact that the solution
contains many multiscale features and requires a very fine mesh to resolve all
details. In order to efficiently compute the solutions, some model reductions
are necessary. To obtain a reduced model, we apply the generalized multiscale
finite element approach, which is a framework allowing systematic construction
of reduced models. Based on this general framework, we will first construct a
local snapshot space, which contains many possible multiscale features of the
solution. Using the snapshot space and a local spectral problem, we identify
dominant modes in the snapshot space and use them as the multiscale basis
functions. Our basis functions are constructed locally with non-overlapping
supports, which enhances the sparsity of the resulting linear system. In order
to enforce the mass conservation, we propose a hybridized technique, and uses a
Lagrange multiplier to achieve mass conservation. We will mathematically
analyze the stability and the convergence of the proposed method. In addition,
we will present some numerical examples to show the performance of the scheme.
We show that, with a few basis functions per coarse region, one can obtain a
solution with excellent accuracy.
