  In this paper, we develop the constrained energy minimizing generalized
multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions
(Dirichlet and Neumann) for the elasticity equations in high contrast media. By
a special treatment of mixed boundary conditions separately, and combining the
construction of the relaxed and constraint version of the CEM-GMsFEM, we
discover that the method offers some advantages such as the independence of the
target region's contrast from precision, while the sizes of oversampling
domains have a significant impact on numerical accuracy. Moreover, to our best
knowledge, this is the first proof of the convergence of the CEM-GMsFEM with
mixed boundary conditions for the elasticity equations given. Some numerical
experiments are provided to demonstrate the method's performance.
