  In this paper, we consider an approximation method, and a novel general
analysis, for second-order elliptic differential equations with heterogeneous
multiscale coefficients. We obtain convergence of the Generalized Multi-scale
Finite Element Method (GMsFEM) method that uses local eigenvectors in its
construction. The analysis presented here can be extended, without great
difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error
estimates generalize and simplify the convergence analysis of [J. Comput. Phys.
230 (2011), 937-955]. The GMsFEM method construct basis functions that are
obtained by multiplication of (approximation of) local eigenvectors by
partition of unity functions. Only important eigenvectors are used in the
construction. The error estimates are general and are written in terms of the
eigenvalues of the eigenvectors not used in the construction. The error
analysis involve local and global norms that measure the decay of the expansion
of the solution in terms of local eigenvectors. Numerical experiments are
carried out to verify the feasibility of the approach with respect to the
convergence and stability properties of the analysis in view of the good
scientific computing practice.
