  In this paper, we address the numerical homogenization approximation of a
free-boundary dam problem posed in a heterogeneous media. More precisely, we
propose a generalized multiscale finite element (GMsFEM) method for the
heterogeneous dam problem. The motivation of using the GMsFEM approach comes
from the multiscale nature of the porous media due to its high-contrast
permeability. Thus, although we can classically formulate the free-boundary dam
problem as in the homogeneous case, a very high resolution will be needed by a
standard finite element approximation in order to obtain realistic results that
recover the multiscale nature. First, we introduce a fictitious time variable
which motivates a suitable time discretization that can be understood as a
fixed point iteration to the steady state solution, and we use a duality method
to deal with the involved multivalued nonlinear terms. Next, we compute
efficient approximations of the pressure and the saturation by using the
GMfsFEM method and we can identify the free boundary. More precisely, the
GMsGEM method provides numerical results that capture the behavior of the
solution due to the variations of the coefficient at the fine-resolution, by
just solving linear systems with size proportional to the number of coarse
blocks of a coarse-grid (that does not need to be adapted to the variations of
the coefficient). Finally, we present illustrative numerical results to
validate the proposed methodology.
