In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lam\'e parameters which are parameterized by a countably infinite number of terms in separate expansions. We estimated the expected values over the infinite dimensional parametric space of linear functionals ${\mathcal L}$ acting on the continuous solution $\vu$ of the elasticity equation. This was achieved by truncating the expansions of the random parameters, then using a high-order quasi-Monte Carlo (QMC) method to approximate the high dimensional integral combined with the conforming Galerkin finite element method (FEM) to approximate the displacement over the physical domain $\Omega.$ In this work, as a further development of aforementioned article, we focus on the case of a nearly incompressible linear stochastic elasticity equation. To serve this purpose, in the presence of stochastic inhomogeneous (variable Lam\'e parameters) nearly compressible material, we develop a new locking-free symmetric nonconforming Galerkin FEM that handles the inhomogeneity. In the case of nearly incompressible material, one known important advantage of nonconforming approximations is that they yield optimal order convergence rates that are uniform in the Poisson coefficient. Proving the convergence of the nonconforming FEM leads to another challenge that is summed up in showing the needed regularity properties of $\vu$. For the error estimates from the high-order QMC method, which is needed to estimate the expected value over the infinite dimensional parametric space of ${\mathcal L}\vu,$ we %rely on (Dick et al. 2022). We are required here to show certain regularity properties of $\vu$ with respect to the random coefficients. Some numerical results are delivered at the end.

翻译：近期工作中(Dick等, arXiv:2310.06187)，我们研究了具有随机Lambé系数的线性随机弹性方程，这些系数通过可数无穷多项的分离展开参数化。我们估计了作用在弹性方程连续解$\vu$上的线性泛函$\mathcal{L}$在无穷维参数空间上的期望值。这一目标的实现通过截断随机参数的展开，随后采用高阶拟蒙特卡洛(QMC)方法逼近高维积分，并结合协调伽辽金有限元方法(FEM)近似物理域$\Omega$上的位移。作为前述文章的进一步发展，本文聚焦于近不可压缩线性随机弹性方程的情形。为此，针对随机非均匀(可变Lambé参数)近不可压缩材料，我们发展了一种新的无锁对称非协调伽辽金有限元方法以处理非均匀性。对于近不可压缩材料，非协调逼近的一个已知重要优势在于其能获得关于泊松系数一致的最优阶收敛速率。证明非协调有限元方法的收敛性带来另一项挑战，其核心在于证明$\vu$所需的正则性性质。对于高阶QMC方法(用于估计无穷维参数空间上$\mathcal{L}\vu$的期望值)的误差估计，我们依赖于(Dick等, 2022)。此处需要证明$\vu$关于随机系数的特定正则性性质。最后给出了若干数值结果。