  In this article we develop the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) for elliptic partial differential
equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions,
and the high contrast property emerges from the coefficients of elliptic
operators and Robin boundary conditions. By careful construction of multiscale
bases of the CEM-GMsFEM, we introduce two operators $\mathcal{D}^m$ and
$\mathcal{N}^m$ which are used to handle inhomogeneous Dirichlet and Neumann
boundary values and are also proved to converge independently of contrast
ratios as enlarging oversampling regions. We provide a priori error estimate
and show that oversampling layers are the key factor in controlling numerical
errors. A series of experiments are conducted, and those results reflect the
reliability of our methods even with high contrast ratios.
