This thesis has seven chapters, the first of which is this introduction. The remainder of the thesis is organized as follows.
– Chapter 2 gives background and introduction to the application area of elliptic and parabolic PDEs, including the particular nonlinear mathematical models that we are considering as model problems in this research.
– In Chapter 3 a review of literature relevant for this thesis is given, including an overview of the computational methods used in the thesis, such as the spatial discretisation, and tem-poral discretisation. We discuss numerical methods such as the Finite Difference Method (FDM) and the family of Backward Differentiation Formulae (BDF) used to approximate
3 1.2. Outline of the Thesis
the model problems. We review the definition and basic properties of nonlinear algebraic equation systems and Differential-Algebraic Equation (DAE) systems.
– In Chapter 4 we introduce several of the most significant methods for solving linear and nonlinear systems of algebraic equations. These include iterative methods for linear sys-tems such as Multigrid and Krylov subspace methods. Numerical algorithms for solutions of nonlinear systems are presented based on the Newton, Newton-Krylov and Multigrid algorithms. Furthermore, we discuss the three nonlinear multilevel algorithms which form the core of this thesis: the FAS multigrid algorithm, Newton-multigrid and the precondi-tioned Newton-Krylov algorithm.
– In Chapter 5 we examine the performance of our numerical implementations of the three nonlinear multilevel algorithms for solving the thin film flow model in 2D. We con-sider these implementations in both steady-state (elliptic) and time-dependent (parabolic) cases. Detailed numerical results and comparison between these algorithms are presented in this chapter as well.
– In Chapter 6 we present the numerical solution of the Cahn-Hilliard-Hele-Shaw (CHHS) system of equations using two nonlinear multilevel algorithms, the FAS and precondi-tioned Newton-Krylov schemes. We present detailed numerical results and make a relative comparison of the approaches.
– In Chapter 7 our conclusions and suggestions for future avenues of research are presented.
– In Appendices A and B, we discuss technical details associated with the nonzero entries in the Jacobian matrix for degrees of freedom next to Dirichlet boundary conditions for the thin film flow model and the CHHS model, respectively. The main body of the thesis only discusses Jacobian entries away from the Dirichlet boundary in order to aid clarity for the reader.
Chapter 2
Mathematical Models
In this chapter, we introduce the mathematical models and concepts that are related to the research in this thesis. In the first section of this chapter, we will present an introduction to the application area of elliptic and parabolic PDEs. Thereafter we discuss certain mathematical models which will subsequently be used to demonstrate and assess the algorithms developed later.
2.1 Introduction to the Application Area of Elliptic and Parabolic PDEs
PDEs are differential equations that include unknown multivariable functions and their asso-ciated partial derivatives. They arise in many areas of science and engineering which govern many natural phenomena. The functions sought have independent variables often representing time and spatial directions. In simple cases, analytical techniques are utilized whereas com-puter algorithms have to be employed for more intricate equations. Let us take a look at a two dimensional, second-order linear partial differential equation which can be written as:
a ∂2u
∂x2 + b ∂2u
∂x∂y + c ∂2u
∂y2 + d ∂u
∂x+ e ∂u
∂y + f u + g = 0, (2.1) according to [87,104,105,121,125]. Here a, b, c, d, e, f, and g may be functions of the independent spatial variables x and y. However for nonlinear problems they may also be functions of the dependent variable u. We can classify PDEs into three broad categories which can be determined by the characteristic polynomial of the highest order derivatives in Equation (2.1). PDEs are elliptic when b2− 4ac < 0, parabolic when b2− 4ac = 0 and hyperbolic when b2− 4ac > 0. We now discuss common examples of the three different types. One of the most common hyperbolic
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5 2.1. Introduction to the Application Area of Elliptic and Parabolic PDEs
PDEs discussed according to [45, 87, 105, 108, 121] is the one-dimensional wave equation:
∂2u
∂t2 = c2 ∂2u
∂x2. (2.2)
Note that, although this is termed ”one-dimensional”, there are still two independent variables.
The reason for this is that in most applications t represents time and x represents the single space variable.
We see in [87, 105, 121] that the simplest example of an elliptic equation is Laplace’s equation:
∂2u
∂x2 + ∂2u
∂y2 = 0, (2.3)
which is generally associated with steady-state behavior in two space dimensions. Furthermore a simple example of a parabolic equation is the one-dimensional heat equation:
∂u
∂t = k ∂2u
∂x2. (2.4)
where u(t, x) signifies the temperature at time t and position x. Usually k is considered con-stant, however, it can depend both on x and t.
In three dimensions, the function u(x, y, z, t) of three spatial variables (x, y, z) and the time variable t, satisfies the heat equation:
Conventionally we can write Equation (2.5) as follows,
∂u
∂t − k ∆u = 0, (2.6)
where ∆ is the Laplace operator for the spatial derivatives.
For a unique solution, we require consistent initial and boundary conditions. Thus we define a bounded region Ω to be the spatial domain while we define (0, T ] to be the temporal region.
Now the full domain for this PDE is Ω× (0, T ] and we require the values of u or its normal derivative at all points to be defined on the boundary of Ω (∂Ω) and the value of u at t = 0 (initial data) at all spatial points. We explore two broad categories of boundary conditions for models in this research: Dirichlet and Neumann. An example, of a Dirichlet boundary condition takes the form:
u = g on ∂ Ω× (0, T ], (2.7)
Chapter 2. Mathematical Models 6
where g is a known function on the boundary ∂ Ω of the spatial region Ω. On the other hand, a Neumann boundary condition takes the form:
∂u
∂ ˜n = f on ∂ Ω× (0, T ], (2.8)
where f is a known function, and ˜n is the outward normal to ∂ Ω. Therefore a positive value of f indicates outflow across the boundary ∂ Ω. As indicated in [32] and [108], PDEs can have purely Neumann boundary conditions or Dirichlet boundary conditions or have a mixture of both. It is possible to define Dirichlet boundary conditions on part of the domain with Neu-mann boundary conditions for the rest.
We have only described linear elliptic, hyperbolic and parabolic PDEs in one, two and three dimensions at this point. The purpose of our work is to study efficient methods to solve non-linear PDEs. Hence the remainder of this section is used to introduce some nonnon-linear PDEs.
In particular, we restrict our attention to nonlinear parabolic and elliptic problems. The inter-ested reader should consult [2,45,87,105,108,121] for discussions on nonlinear hyperbolic PDEs.
There are various forms of nonlinearity to consider. For instance, we can turn the linear Equation (2.4) into a nonlinear equation by defining the coefficient c(u), which is a known nonlinear function, as follows:
∂u
∂t = ∂
∂x
c(u) ∂u
∂x
. (2.9)
The Equation (2.9) is termed the nonlinear diffusion equation (or nonlinear heat equation) in [2, 121], and is an example of a nonlinear parabolic PDE.
We can vary this nonlinear equation in several ways, for example by [72]:
– appending more spatial variables giving,
∂u
∂t = ∂x∂ (c(u) ∂u∂x) +∂y∂ (c(u) ∂u∂y) +∂z∂ (c(u) ∂u∂z), (2.10) where u = u(x, y, z, t),
– adding a nonlinear forcing term f such as
∂u
∂t = ∂x∂ (c(u)∂u∂x) +∂y∂ (c(u)∂u∂y) + f (u, x, y), (2.11) where u = u(x, y, t).
By introducing the divergence operator∇.v which acts on an arbitrary vector function v, and