In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. 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. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.

翻译：本研究探讨了针对随机系数椭圆偏微分方程的非协调间断伽辽金逼近的定制化准蒙特卡洛求积规则开发。我们考虑了输入随机场的仿射均匀模型和对数正态模型，并研究了利用准蒙特卡洛求积规则逼近受输入不确定性影响的偏微分方程响应的期望值。特别地，我们证明：对于间断伽辽金逼近，所得准蒙特卡洛收敛速率与选用连续有限元时表现一致。值得注意的是，本文建立的间断伽辽金参数正则性界同样适用于稀疏网格等其他方法。数值实验结果验证了我们的理论分析。