In this study, we design and develop Quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficient. That is, we consider the affine and uniform, and the lognormal models for the input random field, and investigate the QMC cubatures to compute the response statistics (expectation and variance) of the discretized PDE. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Numerical results underline our analytical findings. Moreover, we present a novel analysis for the parametric regularity in the lognormal setting.

翻译：在此研究中,我们设计并开发了Qasi-Monte Carlo(QMC)幼崽,用于不兼容不连续的 Galerkin(DG) 和随机系数的椭圆部分差异方程近似值。也就是说,我们考虑方形和统一,以及输入随机字段的对数模型,并调查QMC幼崽,以计算离散的PDE的反应统计数据(预测和差异)。特别是,我们证明由此产生的QMC对DG近似值的趋同率与选择连续的有限元素的方式相同。数字结果突出了我们的分析结论。此外,我们提出了对逻辑常态环境中的准度规律性的新分析。