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Mathematics of Computation

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$ H({div})$ preconditioning for a mixed finite element formulation of the diffusion problem with random data

Authors: Howard C. Elman, Darran G. Furnival and Catherine E. Powell
Journal: Math. Comp. 79 (2010), 733-760
MSC (2010): Primary 65F08; Secondary 35R60
Published electronically: December 14, 2009
MathSciNet review: 2600541
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Abstract: We study $ H({div})$ preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for an $ H({div})$ operator with random weight function acting on a certain tensor product space of random fields with finite variance. We build on the Arnold-Falk-Winther multigrid algorithm presented in 1997 by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed. We extend the deterministic analysis to accommodate the modified $ H({div})$ operator and establish spectral equivalence bounds with a new multigrid V-cycle operator that are independent of the spatial and stochastic discretization parameters. We implement multigrid within a block-diagonal preconditioner for the full saddle-point problem, derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned MINRES.

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Additional Information

Howard C. Elman
Affiliation: Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742

Darran G. Furnival
Affiliation: Applied Mathematics and Scientific Computing Program, University of Maryland, College Park, Maryland 20742
Address at time of publication: National Oceanographic Centre, Southampton S014 32H

Catherine E. Powell
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Received by editor(s): June 26, 2008
Received by editor(s) in revised form: February 27, 2009
Published electronically: December 14, 2009
Additional Notes: This work was supported by the U. S. Department of Energy under grant DEFG0204ER25619, and by the U. S. National Science Foundation undergrant CCF0726017.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.