A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
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
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
SIAM Journal on Scientific Computing
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
Stochastic spectral methods for efficient Bayesian solution of inverse problems
Journal of Computational Physics
Domain decomposition for multiscale PDEs
Numerische Mathematik
A multilevel multiscale mimetic (M3) method for two-phase flows in porous media
Journal of Computational Physics
Analysis of FETI methods for multiscale PDEs
Numerische Mathematik
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems
Journal of Computational Physics
Multiscale finite element methods for high-contrast problems using local spectral basis functions
Journal of Computational Physics
Generalized multiscale finite element methods (GMsFEM)
Journal of Computational Physics
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We consider multiscale flow in porous media. We assume that we can characterize the ensemble of all possible flow scenarios, that is, we can describe all possible permeability configurations needed for the simulations. We construct coarse basis functions that can provide inexpensive coarse approximations that are: (1) adequate for all possible flow scenarios in the given ensemble, (2) robust with respect to the small scales and high variations in each flow scenario. The coarse approximations developed here can be used as a multiscale finite element method, or as the coarse solver in a two-level domain decomposition iterative method. The methods presented here extend, to the ensemble case, some of the results in [J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media: reduced dimension coarse spaces, SIAM Multiscale Model. Simul. 8 (5) (2010) 1621-1644] and [Y. Efendiev, J. Galvis, X.H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys. 230 (4) (2011) 937-955]. Specifically, ensembles of permeability fields with high-contrast channels and inclusions are considered. Our main objective here is to construct special multiscale basis functions for the whole ensemble of flow scenarios. The coarse basis functions are pre-computed for (selected or constructed) permeability fields with certain topological properties. This procedure is a preprocessing step and it avoids constructing basis functions (or computing upscaling parameters) for each permeability realization. Then, for any permeability, the solution of elliptic equation can be project to the space spanned by these pre-computed basis functions. We apply this coarse multiscale solver to the design of two-level domain decomposition preconditioner. Numerical experiments show that the ensemble level multiscale finite element method converges to the reference solution. Numerical experiments also show that the ensemble level domain decomposition preconditioner condition number is independent of the high-contrast in the coefficient.