Block-Diagonal Preconditioning for Spectral Stochastic Finite Element Systems

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Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate PDEs whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite element methods (SFEMs) discretise the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy, incorporating only the mean component of the random field coefficient and based on incomplete factorisations has been employed in the literature, and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix and highlight the dependence of the iteration counts on all the SFEM parameters.