ITERATIVE METHODS FOR THE STOCHASTIC DIFFUSION PROBLEM

Postgraduate

ABSTRACT
It is the purpose of this thesis to develop iterative methods for solving the linear systems that arise from application of the stochastic finite element method to steady-state stochastic diffusion problems. Although the theory herein is sufficiently general to be applicable to a variety of choices for the stochastic finite elements, attention is given to the method of polynomial chaos. For the secondorder problem a multigrid algorithm is defined wherein the spatial discretization parameter is varied from grid to grid while the stochastic discretization parameter is held constant. It is demonstrated that the convergence rate of this method is independent of the discretization parameters. For the first-order problem, which produces a linear system that is symmetric and indefinite, the MINRES algorithm is applied with a preconditioner that incorporates a multigrid algorithm.
This multigrid algorithm, as for the one applied to the second-order problem, varies the spatial discretization from grid to grid while holding the stochastic discretization parameter constant. Again, it is demonstrated that the convergence rate of this method is independent of the discretization parameters.

Introduction
It goes without saying that partial differential equations (PDEs) are used ubiquitously in mathematics to model physical phenomena. In this thesis the PDE known as the (steady-state) diffusion equation will be considered. It is given by
    −∇ · (c∇u) = f    inD
where c will be called the diffusion coefficient and f will be called the source function. This is an elliptic PDE and for the existence of solutions should be accompanied by boundary value conditions.
The diffusion equation can be used to model numerous physical processes. Perhaps most simply it can be viewed as giving the steady-state solution of the density of a fluid whose motion has been governed entirely by diffusion (i.e. the flow of fluid from higher density regions to lower density regions). In this context the diffusion equation can be seen to arise from considering conservation of mass with the temporal derivative being omitted as a steady-state solution is sought and the advection term being omitted by assumption. The source function, f, appears as there may be sources (or sinks) of fluid in the spatial domain, D.
Therefore, one sees that the diffusion equation will be incorporated into models of fluid flow through porous media. As discussed, for example, in [11], porous media are inherently heterogeneous and, furthermore, generally defy precise description due to lack of information regarding the properties of the media. Such lack of information may arise from unavoidable ignorance of the system under investigation or the impracticality of understanding the structure of the media at the minute level over which heterogeneity occurs. One therefore sees that the diffusion coefficient can often not be described with certainty. The coarsest approach to resolving this problem would involve setting c: D → R to be the average (or expected value) of the diffusion process. In this case the diffusion coefficient will be called a deterministic field (as its value at each x ∈ D is completely determined, i.e. there is no randomness involved). The model, likewise, will be called the deterministic problem (assuming, of course, that f too is a deterministic field). A more sophisticated approach would be to try and incorporate the uncertainty into the problem by modelling the diffusion coefficient as a random field, i.e. c: D × Ω → R where Ω is a sample space. In this case the resulting problem will be stochastic. In particular, the diffusion equation will now be a stochastic partial differential equation (SPDE) given by
    −∇ · (c∇u) = f    inD × Ω.
The source function too may be a random field. In this thesis, however, the diffusion coefficient will be considered to be the cause of uncertainty. In the succeeding chapters the source function will be considered to be a random field as this is convenient for purposes of analysis and incorporates the fact that f may be deterministic as a special case.
In modelling the random nature of the diffusion coefficient it is generally considered to have known mean, variance, and covariance functions. Such a process will possess a Karhunen-Lo`eve expansion given by
 
where c0 is the mean function of c, (ξr) is a sequence of uncorrelated random variables, and (λr,cr) is a sequence of eigenvalue-eigenvector pairs associated with