  In this paper, we derive an a-posteriori error indicator for the Generalized
Multiscale Finite Element Method (GMsFEM) framework. This error indicator is
further used to develop an adaptive enrichment algorithm for the linear
elliptic equation with multiscale high-contrast coefficients. The GMsFEM, which
has recently been introduced in [12], allows solving multiscale
parameter-dependent problems at a reduced computational cost by constructing a
reduced-order representation of the solution on a coarse grid. The main idea of
the method consists of (1) the construction of snapshot space, (2) the
construction of the offline space, and (3) the construction of the online space
(the latter for parameter-dependent problems). In [12], it was shown that the
GMsFEM provides a flexible tool to solve multiscale problems with a complex
input space by generating appropriate snapshot, offline, and online spaces. In
this paper, we study an adaptive enrichment procedure and derive an
a-posteriori error indicator which gives an estimate of the local error over
coarse grid regions. We consider two kinds of error indicators where one is
based on the $L^2$-norm of the local residual and the other is based on the
weighted $H^{-1}$-norm of the local residual where the weight is related to the
coefficient of the elliptic equation. We show that the use of weighted
$H^{-1}$-norm residual gives a more robust error indicator which works well for
cases with high contrast media. The convergence analysis of the method is
given. In our analysis, we do not consider the error due to the fine-grid
discretization of local problems and only study the errors due to the
enrichment. Numerical results are presented that demonstrate the robustness of
the proposed error indicators.
