We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

中文翻译：

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椭圆型偏微分方程随机均质化的数值研究：代表性体积元大小的收敛速度

我们描述了二维椭圆方程离散化和求解的数值方案，该二维椭圆方程的分段常数系数在多尺度复合材料的随机均质化中会发生很大变化。介绍了一种基于局部Kronecker乘积矩阵组装的高效刚度矩阵生成方案。最终的大型方程组线性系统通过预处理的共轭梯度迭代进行求解，收敛速度与网格大小和跳跃系数（对比度）的变化无关。使用此求解器，我们对随机均质化中代表体积元（RVE）方法的收敛性进行了数值研究，该方法提取了随机系数字段的有效行为。我们的数值实验证实了Gloria等人严格建立的RVE大小的系统误差和标准偏差的渐近收敛速度。还数值研究了均质矩阵以四次张量形式的协方差的渐近行为。我们的方法允许笔记本电脑计算足够大数量的随机实现，即使对于大尺寸的RVE也是如此。