A local-global generalized multiscale finite element method for highly heterogeneous stochastic groundwater flow problems

TL;DR

This paper introduces a novel local-global multiscale finite element method combining reduced basis and GMsFEM to efficiently solve highly heterogeneous stochastic groundwater flow problems with uncertain medium properties.

Contribution

It develops a new hybrid multiscale approach that incorporates local and global information for faster, accurate simulations of stochastic groundwater flow.

Findings

Method achieves higher computational efficiency than individual GMsFEM or reduced basis methods.

Numerical results demonstrate the method's accuracy in modeling heterogeneous media.

The approach effectively reduces the complexity of large linear systems in stochastic simulations.

Abstract

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.