residual method (GMRES) [8,112]. Which method can be used is determined by the properties of the linearised system, which in turn depends on the nonlinear system itself. Here we solve with an iterative Krylov method using GMRES since the Jacobian system is nonsymmetric for the problems we consider.
Krylov-subspace methods are commonly used with a preconditioner that accelerates the con-vergence of the linear iterations. In order to reduce the number of GMRES iterations and speed this algorithm up, several preconditioners have been developed in this research. One important choice of preconditioner is the application of a single V-cycle of multigrid or algebraic multigrid.
This can work well for a problem arising from the discretisation of a single nonlinear elliptic PDE but may not be suited to more general problems [16,47,97,136]. An interested reader may find more details in the literature (cf [9, 16, 30, 42, 78, 82, 98] and [144] amongst many others).
For nonlinear systems, we can consider block-based preconditioners, where multilevel algorithm may be applicable to part of the preconditioning process. Such preconditioners will be designed and discussed as part of this work.
4.4 Summary
This thesis is essentially concerned with the development and application of nonlinear multi-level approaches to the solution of nonlinear elliptic and parabolic systems of PDEs. Therefore, we have presented three common nonlinear multilevel approaches: the first approach is the nonlinear multilevel algorithm (which is the FAS algorithm), then, a linear multigrid method in combination with Newton’s method gives the second approach, and the third multilevel ap-proach is Newton-Krylov with a multilevel preconditioner. In the remainder of this thesis, we consider how to apply these three sophisticated nonlinear multilevel algorithms to nonlinear systems arising from discretising systems of nonlinear PDEs.
All these three approaches can deliver optimal efficiency for certain nonlinear systems as we will see in the later chapters. It is the purpose of this thesis to investigate which is the most efficient. Furthermore, for the Newton-Krylov approach, we will develop novel preconditioners that are suited to problems involving PDE systems.
In Brabazon’s thesis and paper [16, 17] a comparison was made between the Newton-Multigrid and FAS methods for a single PDE. We extend this work to the comparison between these two methods for solving a nonlinear system of PDEs of equations and add the third nonlinear multilevel method for solving the nonlinear system of PDEs.
For the purposes of this thesis, we are interested in implementing these different nonlinear multilevel algorithms for two examples of nonlinear system of PDEs, which will be done in the
Chapter 4. Solution Algorithms for Algebraic Systems 62
later chapters. In the next chapter, we describe their application to the first model in this thesis, that is the thin film flow system of equations introduced in Sections 5.5 and 5.9. After that, we apply them to the second model, namely the Cahn-Hilliard-Hele-Shaw system of equations introduced in Section 6.3.
Chapter 5
Thin Film Flow System
5.1 Introduction
The focus of this chapter is on developing and contrasting efficient and accurate numerical solvers for systems of nonlinear parabolic partial differential equations: specifically, those mod-elling time-dependent thin film flows in two dimensions. The cases of steady-state flows and one-dimensional problems are also considered. A further aim of this chapter is to examine the performance of detailed numerical implementations for three different nonlinear multilevel algo-rithms. We consider these implementations in both steady-state (elliptic) and time-dependent (parabolic) cases, and we make comparisons between them using MATLAB. In all cases, our computational approach uses the finite difference method FDM in space, as we have described in Chapter 3, and implicit time integration with backward differentiation BDF1 in time, which is also described in Chapter 3, for the transient cases.
To solve nonlinear time-dependent problems we must use a nonlinear solver at each time step.
Therefore, at each time step the resulting fully discrete nonlinear algebraic system is solved, with the fsolve MATLAB function in the one-dimensional case, or by using three different non-linear multilevel schemes in the two-dimensional case. The numerical results in 1D and 2D are shown for the computation of steady-state profiles, allowing comparison and validation against previously published work in the one-dimensional case (cf [53,76,145]). We discuss two different mathematical models in the one-dimensional case, as a prelude to the two-dimensional case, selecting the better of these models to then use in two dimensions.
We begin by considering two different mathematical models for the same physical problem 63
Chapter 5. Thin Film Flow System 64
(thin film flow) in the one-dimensional case. We refer to these as Kalliadasis’s model, [76], and Sellier’s model, [51, 53, 145]. Kalliadasis’s model is a third-order PDE given by Equation (2.26), whereas Sellier’s model is given by two coupled second-order PDEs (2.32) and (2.34) in the steady-state case. We approximate both models by using the FDM on regular grids in Section 5.2. In the first instance, we solved both discrete problems using the MATLAB fsolve function. This is used to demonstrate the superiority of Sellier’s scheme hence only this scheme is considered for subsequent investigation. In these investigations, we contrast three different nonlinear multilevel schemes in 2D. Moreover, we have solved Sellier’s model in 2D for both the steady-state and the time-dependent cases with all three of the nonlinear multilevel schemes.
The multilevel schemes considered in this chapter are able to solve a nonlinear algebraic system of equations, arising from discretization of our elliptic and parabolic problems, optimally with a computational expense of O(N ) for N unknowns. We will compute the 2D case with each nonlinear multilevel solution method for both steady-state and time-dependent problems and present their results in sections 5.4 and 5.8. Finally, we will discuss and compare computational performance for all cases individually.