A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
Linear System Theory and Design
Linear System Theory and Design
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
Journal of Computational Physics
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Multiscale finite-volume method for compressible multiphase flow in porous media
Journal of Computational Physics
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations (Lecture Notes in Computational Science and Engineering)
Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation
Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows
SIAM Journal on Scientific Computing
Journal of Computational Physics
Multiscale finite element methods for high-contrast problems using local spectral basis functions
Journal of Computational Physics
Ensemble level multiscale finite element and preconditioner for channelized systems and applications
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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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.