Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs

Abstract

A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and $n$ known, separated microscopic length scales $\epsilon_i$, $i=1,...,n$ in a bounded domain $D\subset R^d$ is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge $P$-a.s, as $\epsilon_i\rightarrow 0$, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension $(n+1)d$. It is shown that this stochastic limit problem admits best $N$-term "polynomial chaos" type approximations which converge at a rate $\sigma>0$ that is determined by the summability of the random inputs' Karhúnen-Loève expansion. The convergence of the polynomial chaos expansion is shown to hold $P$-a.s. and uniformly with respect to the scale parameters $\epsilon_i$. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters $\epsilon_i$ is established in the case of two scales, and in the case of $n>2$ scales convergence is shown, albeit without giving a convergence rate in this case.