Domain decomposition of stochastic PDEs using FEniCS

The intrusive spectral stochastic finite element method (SSFEM) based domain decomposition (DD) solvers for stochastic PDEs are implemented using FEniCS. For efficient uncertainty quantification (UQ) for large- scale computational models, these algorithms demonstrate scalabilities for high resolution spatial discretization and high dimensional random parameter space. However the implementation of these algorithms is intrusive to finite element codes demanding additional programming efforts. In the intrusive SSFEM based DD formalism, the stochastic PDE is converted into a very large set (depending on the number of random parameters) of deterministic coupled PDE system. It leads to a large-scale linear system being solved iteratively using two-level domain decomposition preconditioners. The submatrices for each subdomain are constructed using FEniCS. For each scale of random fluctuation, the associated subdomain level deterministic submatrices required for DD algorithm are extracted through a modified variational form of the PDE. Both three-dimensional stochastic Poisson and linear elasticity problems are tackled through this generic software leveraging FEniCS.