  In this paper, we propose a local-global multiscale method for highly
heterogeneous stochastic groundwater flow problems under the framework of
reduced basis method and the generalized multiscale finite element method
(GMsFEM). Due to incomplete characterization of the medium properties of the
groundwater flow problems, random variables are used to parameterize the
uncertainty. As a result, solving the problem repeatedly is required to obtain
statistical quantities. Besides, the medium properties are usually highly
heterogeneous, which will result in a large linear system that needs to be
solved. Therefore, it is intrinsically inevitable to seek a
computational-efficient model reduction method to overcome the difficulty. We
will explore the combination of the reduced basis method and the GMsFEM. In
particular, we will use residual-driven basis functions, which are key
ingredients in GMsFEM. This local-global multiscale method is more efficient
than applying the GMsFEM or reduced basis method individually. We first
construct parameter-independent multiscale basis functions that include both
local and global information of the permeability fields, and then use these
basis functions to construct several global snapshots and global basis
functions for fast online computation with different parameter inputs. We
provide rigorous analysis of the proposed method and extensive numerical
examples to demonstrate the accuracy and efficiency of the local-global
multiscale method.
