Generalized multiscale finite element methods (GMsFEM)
By Yalchin R. Efendiev, Juan Galvis, Thomas Y. Hou. Journal of Computational Physics, 2013
711 citations recorded by the source corpus.
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.
Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/d61f28e3003ddcf93badff07a356fafbfad46ba7
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In Computational Mechanics
In Computational Theory and Mathematics
In Mechanics of Materials
Citations in the corpus
- Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient (2015, neutral): “The generalized multi-scale finite element method [12] is a systematic method to construct multi-scale basis functions for a family multi-scale problems with parameters.”
- Deep Multiscale Model Learning (2018, background): “Many multiscale methods [23, 22, 38, 33, 21] and solvers are designed to construct coarse spaces and resolve unresolved scales to a desired accuracy via additional computing.”