Generalized Multiscale Finite Element Methods (GMsFEM) (1301.2866v2)

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.

• The paper presents a framework that employs offline-online decomposition to enrich finite element spaces for more accurate multiscale simulations.
• It constructs snapshot spaces refined by spectral decomposition to isolate key solution components, significantly lowering computational complexity.
• Numerical validations in porous media simulations demonstrate the method's robustness and highlight its potential for iterative and adaptive enhancements.

Overview of Generalized Multiscale Finite Element Methods (GMsFEM)

The paper presents a comprehensive framework known as the Generalized Multiscale Finite Element Method (GMsFEM), designed for efficiently simulating multiscale problems amid complex, high-dimensional parameter spaces and without clear scale separation. The method systematically addresses challenges in multiscale simulations by refining solution spaces tailored for local coarse regions, thereby enabling accurate approximations of global solutions.

Key Contributions

1. Framework Development: GMsFEM enhances traditional multiscale finite element methods by incorporating a systematic enrichment process. It decomposes computations into offline and online stages, optimizing efficiency while maintaining accuracy.
2. Snapshot and Offline Spaces: The method begins by constructing snapshot spaces—sets of potential solutions from local problems—which are then refined into online spaces through spectral decomposition. This ensures that only the most influential solution components are retained, significantly reducing computational complexity.
3. Parameter Dependence Handling: GMsFEM is particularly adept at managing input spaces with high dimensionality, such as those encountered in porous media flow simulations. The separation of computation stages ensures the method remains efficient across various permutations of source terms and permeability conditions.
4. Numerical Efficiency: By reducing the problem size only to necessary dimensions, GMsFEM allows for quick adaptation to different input parameters. The reuse of constructed multiscale spaces multiple times reduces overall computation significantly.
5. Numerical Validation: Through extensive numerical examples, the paper validates the method's ability to handle diverse multiscale domains, demonstrating its robustness and adaptability.

Implications and Future Directions

GMsFEM presents significant implications for fields requiring efficient multiscale problem solutions, particularly those facing high computational demands due to intricate input spaces. Potential avenues for future research include further integration with iterative solvers to enhance convergence or exploring adaptive algorithms to optimize the choice of snapshot and offline spaces.

In conclusion, this paper's methodology offers a substantial contribution to the field of computational mathematics, providing a robust tool for efficiently managing multiscale simulations in complex, high-dimensional environments. Future work might focus on extending these concepts to nonlinear and time-dependent problems, presenting a rich area for ongoing exploration and development within numerical analysis and applied mathematics.