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摘要
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 of 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) -%26gt; 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 %26quot;polynomial chaos%26quot; type approximations which converge at a rate sigma %26gt; 0 that is determined by the summability of the random inputs%26apos; Karhunen-Loeve 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 %26gt; 2, scales convergence is shown, albeit without giving a convergence rate in this case.
- 出版日期2013-1
- 单位南阳理工学院
