The black box multigrid numerical homogenization algorithm
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A Simplified Method for Upscaling Composite Materials with High Contrast of the Conductivity
SIAM Journal on Scientific Computing
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In this paper, we develop an upscaling method using coefficient splitting techniques. Green's function is constructed using the differential operator associated with the first part of the splitting. An effective upscaling coefficient is recursively calculated by Green's function. The computation of the upscaling process involves some independent steps. Combining the proposed upscaling method with the stochastic collocation method, we present a stochastic space reduction collocation method, where the stochastic collocation method is performed on a lower dimension stochastic space than the full-dimension stochastic space. We thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic PDEs. Computation complexity is also addressed for the stochastic upscaling method. A number of numerical tests are presented to confirm the convergence analysis.