Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Mixed Finite Element Methods on Nonmatching Multiblock Grids
SIAM Journal on Numerical Analysis
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Analysis of a Two-Scale, Locally Conservative Subgrid Upscaling for Elliptic Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A Posteriori Error Estimates for the Mortar Mixed Finite Element Method
SIAM Journal on Numerical Analysis
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
A stochastic variational multiscale method for diffusion in heterogeneous random media
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Parallel Domain Decomposition Methods for Stochastic Elliptic Equations
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows
SIAM Journal on Scientific Computing
Numerical Approximation of Partial Differential Equations
Numerical Approximation of Partial Differential Equations
Hi-index | 0.00 |
This paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcy's law with nonstationary stochastic permeability represented as a sum of local Karhunen-Loève expansions. The approximation uses stochastic collocation on either a tensor product or a sparse grid, coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method. The latter method requires solving a coarse scale mortar interface problem via an iterative procedure. The traditional implementation requires the solution of local fine scale linear systems on each iteration. We employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration. In the stochastic setting, the basis is further reused over multiple realizations, leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude. Error analysis and numerical experiments are presented.