Residual-driven online generalized multiscale finite element methods

@article{Chung2015ResidualdrivenOG,
  title={Residual-driven online generalized multiscale finite element methods},
  author={Eric T. Chung and Yalchin R. Efendiev and Wing Tat Leung},
  journal={J. Comput. Phys.},
  year={2015},
  volume={302},
  pages={176-190}
}
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85 Citations
Background Citations
30
Methods Citations
34
Results Citations
2

85 Citations

Fast online generalized multiscale finite element method using constraint energy minimization
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TLDR
A general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition and coupling using Hybridized Discontinuous Galerkin is proposed, which shows that one can achieve a fast convergence when using online basis functions.
Adaptive generalized multiscale approximation of a mixed finite element method with velocity elimination
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The numerical results show that with a suitable initial multiscale space that includes all offline basis functions corresponding to relative smaller eigenvalues of each local spectral decomposition in the offline stage, the convergence rate of the online enrichment is independent of the permeability contrast.
An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media
TLDR
An online local adaptivity technique for local multiscale model reduction problems based on local residuals and some optimally estimates is developed and it is shown that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully.
Sparse Generalized Multiscale Finite Element Methods and their applications
TLDR
This paper uses local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space, and directly applies these techniques to solve the underlying PDEs.
Residual driven online mortar mixed finite element methods and applications
Online adaptive algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method
TLDR
An online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method (CEM-GMsDGM), which shows the proposed online enrichment leads to a fast convergence from multiscales approximation to the fine-scale solution.
Cluster-based generalized multiscale finite element method for elliptic PDEs with random coefficients
Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media
TLDR
The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed.
Re-iterated multiscale model reduction using the GMsFEM
Numerical homogenization and multiscale finite element methods construct effective properties on a coarse grid by solving local problems and extracting the average effective properties from these…
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References

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The motivation stems from the analysis presented in this paper, which show that when using oversampling techniques in the construction of the snapshot space and offline space, GMsFEM will converge independent of small scales and high-contrast under certain assumptions.
An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media
TLDR
An online local adaptivity technique for local multiscale model reduction problems based on local residuals and some optimally estimates is developed and it is shown that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully.
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