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IMA Journal of Numerical Analysis Advance Access originally published online on April 4, 2008
IMA Journal of Numerical Analysis 2009 29(2):350-375; doi:10.1093/imanum/drn014
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Catherine E. Powell{dagger}

School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

Howard C. Elman{ddagger}

Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA

{dagger} Corresponding author. Email: c.powell{at}manchester.ac.uk

{ddagger} Email:elman{at}cs.umd.edu

Received on 25 May 2007. Revised on 9 January 2008.


  Abstract

Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

Key Words: finite elements; stochastic finite elements; fast solvers; preconditioning; multigrid


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