  In this paper, we study the generalized multiscale finite element method
(GMsFEM) for single phase compressible flow in highly heterogeneous porous
media. We follow the major steps of the GMsFEM to construct permeability
dependent offline basis for fast coarse-grid simulation. The offline coarse
space is efficiently constructed only once based on the initial permeability
field with parallel computing. A rigorous convergence analysis is performed for
two types of snapshot spaces. The analysis indicates that the convergence rates
of the proposed multiscale method depend on the coarse meshsize and the
eigenvalue decay of the local spectral problem. To further increase the
accuracy of multiscale method, residual driven online multiscale basis is added
to the offline space. The construction of online multiscale basis is based on a
carefully design error indicator motivated by the analysis. We find that online
basis is particularly important for the singular source. Rich numerical tests
on typical 3D highly heterogeneous medias are presented to demonstrate the
impressive computational advantages of the proposed multiscale method.
