Polynomial approximations of a class of stochastic multiscale elasticity problems

Abstract

We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic moduli depend on $n$ separated microscopic scales, are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger-Reissner problem that allows for computing the stress directly, and the multiscale mixed problem for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information, whose solutions are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of $N$ gpc modes. We deduce bounds and summability properties for the solutions' gpc expansion coefficients, which imply explicit rates of convergence in terms of $N$ when the gpc modes used for the Galerkin approximation are chosen to correspond to the best $N$ terms in the gpc expansion. For the mixed problem for nearly incompressible materials, the rate of convergence for the best $N$ term approximation is independent of the Lam\'e constants' ratio. We establish parametric correctors in terms of the semidiscrete Galerkin approximations. For two scale problems, an explicit homogenization rate is deduced. Together with the best $N$ term rate, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization rate that is independent of the ratio of the Lam\'e constants, so that the error for the corrector is also independent of this ratio.