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
Analysis of a Two-Scale, Locally Conservative Subgrid Upscaling for Elliptic Problems
SIAM Journal on Numerical Analysis
Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Journal of Computational Physics
A stochastic variational multiscale method for diffusion in heterogeneous random media
Journal of Computational Physics
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A stochastic multiscale framework for modeling flow through random heterogeneous porous media
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Kernel principal component analysis for stochastic input model generation
Journal of Computational Physics
Kernel principal component analysis for stochastic input model generation
Journal of Computational Physics
A probabilistic graphical model approach to stochastic multiscale partial differential equations
Journal of Computational Physics
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A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen-Loeve expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.