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

Elman, Howard; Furnival, Darran; Powell, Catherine

doi:10.1090/S0025-5718-09-02274-1

$H(\mathrm {div})$ 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;

Math. Comp. 79 (2010), 733-760

DOI: https://doi.org/10.1090/S0025-5718-09-02274-1

Published electronically: December 14, 2009

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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.

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