Variational Multiscale Analysis: The Fine-Scale Green's Function for Stochastic Partial Differential Equations

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) we 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.