LOCALIZED STOCHASTIC GALERKIN METHODS FOR HELMHOLTZ PROBLEMS CLOSE TO RESONANCE

Efficiently solving stochastic Helmholtz equations remains an open challenging problem, especially when the corresponding problems are close to resonant frequencies. For widely used stochastic Galerkin methods based on spectral stochastic finite element approximations, two main computational difficulties exist when solving this kind of problem: slow convergence rates of spectral approximation methods and efficiency degeneration of preconditioned iterative linear solvers. To address this issue, we focus on the multi-element generalized polynomial chaos (ME-gPC) for stochastic approximation and finite element methods for physical approximation. A novel localized stochastic Galerkin scheme based on the combination of ME-gPC finite element approximation and mean-based preconditioning is proposed and analyzed in this work. Theoretical analysis shows that the mean-based preconditioner can be efficient in this setting, and numerical studies demonstrate the overall efficiency of the localized stochastic Galerkin scheme to solve the stochastic Helmholtz equations close to resonance.