  In this paper we develop two goal-oriented adaptive strategies for a
posteriori error estimation within the generalized multiscale finite element
framework. In this methodology, one seeks to determine the number of multiscale
basis functions adaptively for each coarse region to efficiently reduce the
error in the goal functional. Our first error estimator uses a residual based
strategy where local indicators on each coarse neighborhood are the product of
local indicators for the primal and dual problems, respectively. In the second
approach, viewed as the multiscale extension of the dual weighted residual
method (DWR), the error indicators are computed as the pairing of the local
H^{-1} residual of the primal problem weighed by a projection into the primal
space of the H_0^1 dual solution from an enriched space, over each coarse
neighborhood. In both of these strategies, the goal-oriented indicators are
then used in place of a standard residual-based indicator to mark coarse
neighborhoods of the mesh for further enrichment in the form of additional
multiscale basis functions. The method is demonstrated on high-contrast
problems with heterogeneous multiscale coefficients, and is seen to outperform
the standard residual based strategy with respect to efficient reduction of
error in the goal function.
