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- W2952157049 abstract "We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect of the unresolved fine scales through a model term that contains a fine-scale stochastic Green's function. For a natural choice of an optimal coarse-scale solution and L^2-orthogonal stochastic basis functions, we demonstrate that the fine-scale stochastic Green's function is intimately linked to its deterministic counterpart. In particular, (i) we demonstrate that whenever the deterministic fine-scale function vanishes, the stochastic fine-scale function satisfies a weaker, and discrete notion of vanishing stochastic coefficients, and (ii) derive an explicit formula for the fine-scale stochastic Green's function that only involves quantities needed to evaluate the fine-scale deterministic Green's function. We present numerical results that support our claims about the physical support of the stochastic fine-scale function, and demonstrate the benefit of using the VMS method when the fine-scale Green's function is approximated by an easier to implement, element Green's function." @default.
- W2952157049 created "2019-06-27" @default.
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- W2952157049 date "2013-10-07" @default.
- W2952157049 modified "2023-09-27" @default.
- W2952157049 title "Variational multiscale analysis: the fine-scale Green's function for stochastic partial differential equations" @default.
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