Generalized Multiscale Finite Element Methods (GMsFEM)

TL;DR

This paper introduces GMsFEM, a flexible multiscale simulation method that efficiently constructs local solution spaces for complex problems without scale separation, enabling fast online computations and reusability across different inputs.

Contribution

The paper presents a systematic offline-online framework for GMsFEM, including a spectral decomposition-based enrichment process that enhances multiscale solution accuracy efficiently.

Findings

Effective reduction of degrees of freedom in multiscale simulations.

Fast online construction of multiscale spaces for various inputs.

Numerical results demonstrating high accuracy and computational savings.

Abstract

In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.