Preconditioning Stochastic Galerkin Methods of Diffusion Problems with Random Data
When solving stochastic partial differential equations with random coefficients, the stochastic Galerkin method results in a large single system via a traditional finite element discretization in the spatial domain and generalized polynomial chaos in the stochastic space. The sparse pattern on the stochastic Galerkin matrix can be described by a simplex lattice structure. With the help of this simplex structure, it is shown that the spectrum of the mean-based preconditioning system is symmetric with respect to one and a tighter eigen-bound for the mean-based preconditioned system is developed. Furthermore, a new variance-involved block diagonal preconditioner and a new block triangular preconditioner based on the simplex lattice structure are proposed and their corresponding spectral analyses are studied. Numerical experiments show that the new preconditioners are more robust and efficient compared to the mean-based preconditioner, especially when the variance and the total polynomial degree in the complete stochastic space are large. The block triangular preconditioner is extended to a stochastic diffusion problem with mixed formulation together with a new mean-based block triangular preconditioner. Both preconditioners outperform the traditional mean-based diagonal preconditioner on a benchmark problem.