  Complex processes in perforated domains occur in many real-world
applications. These problems are typically characterized by physical processes
in domains with multiple scales (see Figure 1 for the illustration of a
perforated domain). Moreover, these problems are intrinsically multiscale and
their discretizations can yield very large linear or nonlinear systems. In this
paper, we investigate multiscale approaches that attempt to solve such problems
on a coarse grid by constructing multiscale basis functions in each coarse
grid, where the coarse grid can contain many perforations. In particular, we
are interested in cases when there is no scale separation and the perforations
can have different sizes. In this regard, we mention some earlier pioneering
works [14, 18, 17], where the authors develop multiscale finite element
methods. In our paper, we follow Generalized Multiscale Finite Element Method
(GMsFEM) and develop a multiscale procedure where we identify multiscale basis
functions in each coarse block using snapshot space and local spectral
problems. We show that with a few basis functions in each coarse block, one can
accurately approximate the solution, where each coarse block can contain many
small inclusions. We apply our general concept to (1) Laplace equation in
perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes
equations in perforated domain. Numerical results are presented for these
problems using two types of heterogeneous perforated domains. The analysis of
the proposed methods will be presented elsewhere.
