Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for instance, arising in linear elasticity, the existing two-level domain decomposition solvers with the vertex-based coarse grid show poor numerical and parallel scalabilities. Therefore, new algorithms with a better resolved coarse grid are needed. The probabilistic wirebasket-based coarse grid for a two-level solver is devised in three dimensions. This enriched coarse grid provides an efficient mechanism for global error propagation and thus improves the convergence. This development enhances the scalability of the two-level solver in handling stochastic PDEs in three dimensions. Numerical and parallel scalabilities of this algorithm are studied using MPI and PETSc libraries on high-performance computing (HPC) systems. Implementational challenges of the intrusive spectral stochastic finite element methods (SSFEM) are addressed by coupling domain decomposition solvers with FEniCS general purpose finite element package. This work generalizes the applications of intrusive SSFEM to tackle a variety of stochastic PDEs and emphasize the usefulness of the domain decomposition-based solvers and HPC for uncertainty quantification.
翻译:在输入和输出过程中,现实物理现象在多个尺度上表现出随机波动。这些现象的模型要求有随机的 PDEs 。例如,对于三维结合(Vctor-value)随机的 PDEs (SPDEs) 来说,在线性弹性方面,现有的双级域分解解解解解解解解溶器与顶端粗格网格相比,其数值和平行的伸缩性都显示低。因此,需要使用基于高性能计算系统的MPI和PETSc图书馆来研究这种算法的数值和平行的伸缩性。2级求解解密的铁篮子网以三维方式设计出一个双级求解析网格。这种浓缩的网格为全球错误传播提供了一个有效的机制,从而改进了趋同性PDESDS(SSD) 提供了一种可控频谱谱分解的磁性磁性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性能能能能能能能能能能能能能能能能能能能能能能能