An upscaling method using coefficient splitting and its applications to elliptic PDEs

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.