  In this paper, we propose offline and online adaptive enrichment algorithms
for the generalized multiscale approximation of a mixed finite element method
with velocity elimination to solve the subsurface flow problem in high-contrast
and heterogeneous porous media. We give the theoretical analysis for the
convergence of these two adaptive methods, which shows that sufficient initial
basis functions (belong to the offline space) leads to a faster convergence
rate. A series of numerical examples are provided to highlight the performance
of both these two adaptive methods and also validate the theoretical analysis.
Both offline and online adaptive methods are effective that can reduce the
relative error substantially. In addition, the online adaptive method generally
performs better than the offline adaptive method as online basis functions
contain important global information such as distant effects that cannot be
captured by offline basis functions. The numerical results also show that with
a suitable initial multiscale space that includes all offline basis functions
corresponding to relative smaller eigenvalues of local spectral decompositions
in the offline stage, the convergence rate of the online enrichment is
independent of the permeability contrast.
