  This work is concerned with the rigorous analysis on the Generalized
Multiscale Finite Element Methods (GMsFEMs) for elliptic problems with
high-contrast heterogeneous coefficients. GMsFEMs are popular numerical methods
for solving flow problems with heterogeneous high-contrast coefficients, and it
has demonstrated extremely promising numerical results for a wide range of
applications. However, the mathematical justification of the efficiency of the
method is still largely missing.
  In this work, we analyze two types of multiscale basis functions, i.e., local
spectral basis functions and basis functions of local harmonic extension type,
within the GMsFEM framework. These constructions have found many applications
in the past few years. We establish their optimal convergence in the energy
norm under a very mild assumption that the source term belongs to some weighted
$L^2$ space, and without the help of any oversampling technique. Furthermore,
we analyze the model order reduction of the local harmonic extension basis and
prove its convergence in the energy norm. These theoretical findings shed
insights into the mechanism behind the efficiency of the GMsFEMs.
