  Numerical homogenization and multiscale finite element methods construct
effective properties on a coarse grid by solving local problems and extracting
the average effective properties from these local solutions. In some cases, the
solutions of local problems can be expensive to compute due to scale disparity.
In this setting, one can basically apply a homogenization or multiscale method
re-iteratively to solve for the local problems. This process is known as
re-iterated homogenization and has many variations in the numerical context.
Though the process seems to be a straightforward extension of two-level
process, it requires some careful implementation and the concept development
for problems without scale separation and high contrast. In this paper, we
consider the Generalized Multiscale Finite Element Method (GMsFEM) and apply it
iteratively to construct its multiscale basis functions. The main idea of the
GMsFEM is to construct snapshot functions and then extract multiscale basis
functions (called offline space) using local spectral decompositions in the
snapshot spaces. The extension of this construction to several levels uses
snapshots and offline spaces interchangebly to achieve this goal. At each
coarse-grid scale, we assume that the offline space is a good approximation of
the solution and use all possible offline functions or randomization as
boundary conditions and solve the local problems in the offline space at the
previous (finer) level, to construct snapshot space. We present an adaptivity
strategy and show numerical results for flows in heterogeneous media and in
perforated domains.
