针对近不可压缩平面随机弹性方程的高阶QMC非协调有限元方法
High-order QMC nonconforming FEMs for nearly incompressible planar stochastic elasticity equations
AI总结：
本文针对近不可压缩平面随机弹性方程，提出了一种处理随机非均匀材料的免锁定对称非协调Galerkin FEM，并结合高阶QMC方法估计无限维参数空间上的期望值，证明了所需解的正则性并给出了数值结果。
摘要：
在最近的工作（Dick et al, arXiv:2310.06187）中，我们考虑了一个具有随机 Lamé 参数的线性随机弹性方程，这些参数通过可数无限项的独立展开进行参数化。我们估计了作用在弹性方程连续解 $\vu$ 上的线性泛函 $\mathcal{L}$ 在无限维参数空间上的期望值。这是通过截断随机参数的展开，然后使用高阶准蒙特卡洛（QMC）方法来近似高维积分，并结合协调 Galerkin 有限元方法（FEM）来近似物理域 $\Omega$ 上的位移来实现的。在这项工作中，作为前述文章的进一步发展，我们关注近不可压缩线性随机弹性方程的情况。为此，在存在随机非均匀（可变 Lamé 参数）近可压缩材料的情况下，我们开发了一种新的免锁定对称非协调 Galerkin FEM，用于处理非均匀性。在近不可压缩材料的情况下，非协调近似的一个已知重要优势是它们产生在 Poisson 系数上一致的最优阶收敛率。证明非协调 FEM 的收敛性带来了另一个挑战，即证明 $\vu$ 所需的正则性性质。对于高阶 QMC 方法的误差估计（这是估计 $\mathcal{L}\vu$ 在无限维参数空间上期望值所必需的），我们依赖于（Dick et al. 2022）。这里我们需要证明 $\vu$ 关于随机系数的某些正则性性质。最后给出了一些数值结果。
英文摘要：
In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lamé 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 $Ω.$ 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 Lamé 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.
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