  This work presents an Iterative Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (ICEM-GMsFEM) for solving the contact problem
with high contrast coefficients. The model problem can be characterized by a
variational inequality, where we add a penalty term to convert this problem
into a non-smooth and non-linear unconstrained minimizing problem. The
characterization of the minimizer satisfies the variational form of a mixed
Dirilect-Neumann-Robin boundary value problem. So we apply CEM-GMsFEM
iteratively and introduce special boundary correctors along with multiscale
spaces to achieve an optimal convergence rate. Numerical results are conducted
for different highly heterogeneous permeability fields, validating the fast
convergence of the CEM-GMsFEM iteration in handling the contact boundary and
illustrating the stability of the proposed method with different sets of
parameters. We also prove the fast convergence of the proposed iterative
CEM-GMsFEM method and provide an error estimate of the multiscale solution
under a mild assumption.
