$H(\mathrm {div})$ preconditioning for a mixed finite element formulation of the diffusion problem with random data
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- by Howard C. Elman, Darran G. Furnival and Catherine E. Powell PDF
- Math. Comp. 79 (2010), 733-760 Request permission
Abstract:
We study $H(\mathrm {div})$ 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 $H(\mathrm {div})$ 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 $H(\mathrm {div})$ 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.References
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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
- Email: elman@cs.umd.edu
- 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
- Email: d.furnival@noc.soton.ac.uk
- Catherine E. Powell
- Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
- MR Author ID: 739396
- Email: c.powell@manchester.ac.uk
- 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.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 733-760
- MSC (2010): Primary 65F08; Secondary 35R60
- DOI: https://doi.org/10.1090/S0025-5718-09-02274-1
- MathSciNet review: 2600541