# Stochastic Diffusion Problem: Iterative Methods

The main objective in this thesis was to create iterative methods for solving the linear systems which appear from using the stochastic finite element approach to steady-state stochastic diffusion problems. While the theory here is standard and it can be relevant to a number of choices for the stochastic finite elements, consideration is given to the method of polynomial chaos. For the second order problem a multigrid algorithm is described in which the spatial discretization parameter is varied from grid to grid while the stochastic discretization parameter is kept constant. It is revealed that the convergence rate of this technique is independent of the discretization parameters. We have applied the MINRES algorithm which incorporates a multigrid algorithm for the first-order problem, it produces a linear system that is symmetric and indefinite. This multigrid algorithm which was applied to the second-order problem….

## Contents: Stochastic Diffusion Problem

1 Introduction
2 Second-Order Stochastic Diffusion Problem
2.1 Boundary Value Problem
2.2 Lebesgue and Sobolev Spaces
2.2.1 Spaces L2(D), L2(
), and L2(D) ⊗ L2(
)
2.2.2 Spaces L∞(D), L∞(
), and L∞(D) ⊗ L∞(
)
2.2.3 Spaces Hk(D) and Hk(D) ⊗ L2(D)
2.3 Weak Formulation
2.4 Stochastic Finite Element Method
2.5 Deterministic Finite Elements
2.6 Polynomial Chaos
2.6.1 Hermite Polynomials
2.6.2 Legendre Polynomials
2.7 Matrix Formulation
2.7.1 Post-processing the Matrix Solution
2.8 Model Problem
2.9 Matrix and Right Hand Side Properties
2.10 Semi-discrete Finite Element Formulation
3 Solving the Second-Order Stochastic Diffusion Problem
3.1 Stationary Iteration
3.2 Two-grid Correction Scheme
3.3 Convergence of Two-Grid Correction Scheme
3.4 Smoothing Property
3.5 Approximation Property
3.6 Extension to Multigrid
3.7 Numerical Experiments
3.7.1 Diffusion with Uniform Distributions
3.7.2 Diffusion with Normal Distributions
4 First-Order Stochastic Diffusion Problem
4.1 Boundary Value Problem
4.2 Lebesgue and Sobolev Spaces
4.2.1 Spaces Hk(D)2 and Hk(D)2 ⊗ L2(
)
4.2.2 Spaces H(div;D) and H(div;D) ⊗ L2(
)
4.3 Weak Formulation
4.4 Mixed Stochastic Finite Element Method
4.5 Matrix Formulation
4.6 Model Problem
4.7 Weighted H(div;D) ⊗ L2(
) Bilinear Form
4.8 Raviart-Thomas Interpolation Operator
4.9 Semi-discrete Mixed Finite Element Formulation
4.10 Helmholtz Decomposition
4.11 Projection Operators
4.12 Weighted H(div;D) ⊗ L2(
) Operator
5 Solving the First-Order Stochastic Diffusion Problem
5.1 Deterministic H(div;D) Preconditioner
5.2 Stochastic H(div;D) ⊗ L2(
) Preconditioner
5.3 Two-grid Function Bounds
5.4 Λ-Projection Bounds
5.5 Additive Schwarz Method
5.6 Multigrid
5.7 Numerical Experiments
5.7.1 Diffusion with Uniform Distributions………