  The main goal of this paper is to design multiscale basis functions within
GMsFEM framework such that the convergence of method is independent of the
contrast and linearly decreases with respect to mesh size if oversampling size
is appropriately chosen. We would like to show a mesh-dependent convergence
with a minimal number of basis functions. Our construction starts with an
auxiliary multiscale space by solving local spectral problems. In auxiliary
multiscale space, we select the basis functions that correspond to small
(contrast-dependent) eigenvalues. These basis functions represent the channels
(high-contrast features that connect the boundaries of the coarse block). Using
the auxiliary space, we propose a constraint energy minimization to construct
multiscale spaces. The minimization is performed in the oversampling domain,
which is larger than the target coarse block. The constraints allow handling
non-decaying components of the local minimizers. If the auxiliary space is
correctly chosen, we show that the convergence rate is independent of the
contrast (because the basis representing the channels are included in the
auxiliary space) and is proportional to the coarse-mesh size (because the
constrains handle non-decaying components of the local minimizers). The
oversampling size weakly depends on the contrast as our analysis shows. The
convergence theorem requires that channels are not aligned with the coarse
edges, which hold in many applications, where the channels are oblique with
respect to the coarse-mesh geometry. The numerical results confirm our
theoretical results. In particular, we show that if the oversampling domain
size is not sufficiently large, the errors are large. To remove the
contrast-dependence of the oversampling size, we propose a modified
construction for basis functions and present numerical results and the
analysis.
