Multilevel Numerical Algorithms for Systems of Nonlinear Parabolic Partial Differential Equations

This thesis is concerned with the development of efficient and reliable numerical algorithms for the solution of nonlinear systems of partial differential equations (PDEs) of elliptic and parabolic type. The main focus is on the implementation and performance of three different nonlinear multilevel algorithms, following discretisation of the PDEs: the Full Approximation Scheme (FAS), Newton-Multigrid (Newton-MG) and a Newton-Krylov solver with a novel pre- conditioner that we have developed based on the use of Algebraic Multigrid (AMG). In recent years these algorithms have been commonly used to solve nonlinear systems that arise from the discretisation of PDEs due to the fact that their execution time can scale linearly (or close to linearly) with the number of degrees of freedom used in the discretisation.
We consider two mathematical models: a thin film flow and the Cahn-Hilliard-Hele-Shaw model. These mathematical models consist of nonlinear, time-dependent and coupled PDEs systems. Using a Finite Difference Method (FDM) in space and Backward Differentiation For- mulae (BDF) in time, we discrete the two models, to produce nonlinear algebraic systems. We are able to solve these nonlinear systems implicitly in computationally demanding 2D situa- tions. We present numerical results, for both steady-state and time-dependent problems, that demonstrate the optimality of the three numerical algorithms for the thin film flow model. We show optimality of the FAS and Newton-Krylov approaches for the time-dependent Cahn- Hilliard-Hele-Shaw (CHHS) problem.
The main contribution is to address the question of which of these three nonlinear solvers is likely to be the best (i.e. computationally most effective) in practice. In order to asses this, we discuss the careful implementation and timing of these algorithms in order to permit a fair direct comparison of their computational cost. We then present extensive numerical results in order to make this comparison between these nonlinear multilevel methods. The conclusion emerging from this investigation is that it does not appear that there is a single superior approach, but rather that the best approach is problem dependent. Specifically, we find that our optimally preconditioned Newton-Krylov approach is best for the thin film flow model in the steady-state and time-dependent form, whilst the FAS solver appears best for the time-dependent CHHS model.