H(div) Preconditioning for a Mixed Finite Element Formulation of the Stochastic Diffusion Problem

Furnival, Darran and Elman, Howard and Powell, Catherine (2008) H(div) Preconditioning for a Mixed Finite Element Formulation of the Stochastic Diffusion Problem. [MIMS Preprint]

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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 a weighted, stochastic H(div) operator, acting on a certain tensor product space of random fields with finite variance. We build on the Arnold-Falk-Winther multigrid algorithm presented in \cite{arnold1997} 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 stochastic 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.

Item Type: MIMS Preprint
Uncontrolled Keywords: stochastic finite elements, multigrid, H(div) operator
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr C.E. Powell
Date Deposited: 27 Jun 2008
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1119

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