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Keywords:

stochastic finite element method, polynomial chaos, multigrid, preconditioning, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, PARTIAL-DIFFERENTIAL-EQUATIONS, CIRCULANT PRECONDITIONERS, POLYNOMIAL CHAOS, SYSTEMS, COEFFICIENTS, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0802 Computation Theory and Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics, 4903 Numerical and computational mathematics

Abstract:

This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs . Specialized solvers are required to solve the very high-dimensional systems that result after a finite e lement discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; a nd multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Mult ilevel methods based on coarsening the stochastic dimension perform in general poorly due to a lar ge computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based precon- ditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties. © 2010 Society for Industrial and Applied Mathematics.