Recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions. A two-level domain decomposition method is reported in this investigation to devise a linear solver for the large-scale system in the Galerkin spectral stochastic finite element method (SSFEM). In particular, a two-level scalable preconditioner is introduced in order to iteratively solve the large-scale linear system in the intrusive SSFEM using an iterative substructuring based domain decomposition solver. The implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains. The numerical and parallel scalabilities of the two-level preconditioner are contrasted with the previously developed one-level preconditioner for two-dimensional flow through porous media and elasticity problems with spatially varying non-Gaussian material properties. A distributed implementation of the parallel algorithm is carried out using MPI and PETSc parallel libraries. The scalabilities of the algorithm are investigated in a Linux cluster.

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Keywords Balancing domain decomposition by constraints, Domain decomposition method, Neumann-Neumann preconditioner, Polynomial chaos expansion, Schur complement system, Stochastic finite element method, Stochastic PDEs
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Journal Journal of Computational Physics
Subber, W. (Waad), & Sarkar, A. (2014). A domain decomposition method of stochastic PDEs: An iterative solution techniques using a two-level scalable preconditioner. Journal of Computational Physics, 257(PA), 298–317. doi:10.1016/