  In this paper, we present two adaptive methods for the basis enrichment of
the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the
flow problem in heterogeneous media. We develop an a-posteriori error indicator
which depends on the norm of a local residual operator. Based on this
indicator, we construct an offline adaptive method to increase the number of
basis functions locally in coarse regions with large local residuals. We also
develop an online adaptive method which iteratively enriches the function space
by adding new functions computed based on the residual of the previous solution
and special minimum energy snapshots. We show theoretically and numerically the
convergence of the two methods. The online method is, in general, better than
the offline method as the online method is able to capture distant effects (at
a cost of online computations), and both methods have faster convergence than a
uniform enrichment. Analysis shows that the online method should start with
certain number of initial basis functions in order to have the best
performance. The numerical results confirm this and show further that with
correct selection of initial basis functions, the convergence of the online
method can be independent of the contrast of the medium. We consider cases with
both very high and very low conducting inclusions and channels in our numerical
experiments.
