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SIAM J. on Scientific Computing / Volume 32 / Issue 1

SIAM J. Sci. Comput. 32, pp. 372-397 (26 pages)

Iterative Solvers for the Stochastic Finite Element Method

Eveline Rosseel and Stefan Vandewalle

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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 element 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; and 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. Multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based preconditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties.

© 2010 Society for Industrial and Applied Mathematics

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KEYWORDS

AMS Subject Headings

65N22, 35R60, 65F10

PUBLICATION DATA

ISSN

1064-8275 (print)  
1095-7197 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received June 12, 2008
Accepted June 04, 2009
Published online February 05, 2010
Digital Object Identifier
http://dx.doi.org/10.1137/080727026

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