Schwarz preconditioners for stochastic elliptic PDEs

Increasingly the spectral stochastic finite element method (SSFEM) has become a popular computational tool for uncertainty quantification in numerous practical engineering problems. For large-scale problems however, the computational cost associated with solving the arising linear system in the SSFEM still poses a significant challenge. The development of efficient and robust preconditioned iterative solvers for the SSFEM linear system is thus of paramount importance for uncertainty quantification of large-scale industrially relevant problems. In the context of high performance computing, the preconditioner must scale to a large number of processors. Therefore in this paper, a two-level additive Schwarz preconditioner is described for the iterative solution of the SSFEM linear system. The proposed preconditioner can be viewed as a generalization of the mean based block-diagonal preconditioner commonly used in the literature. For the numerical illustrations, two-dimensional steady-state diffusion and elasticity problems with spatially varying random coefficients are considered. The performance of the algorithm is investigated with respect to the geometric parameters, strength of randomness, dimension and order of the stochastic expansion.