White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Multigrid methods for nonlinear second order partial differential operators

Brabazon, Keeran J (2014) Multigrid methods for nonlinear second order partial differential operators. PhD thesis, University of Leeds.

[img]
Preview
Text (PhD Thesis)
thesis.pdf - Final eThesis - complete (pdf)
Available under License Creative Commons Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales.

Download (1245Kb) | Preview

Abstract

This thesis is concerned with the efficient numerical solution of nonlinear partial differential equations (PDEs) of elliptic and parabolic type. Such PDEs arise frequently in models used to describe many physical phenomena, from the diffusion of a toxin in soil to the flow of viscous fluids. The main focus of this research is to better understand the implementation and performance of nonlinear multigrid methods for the solution of elliptic and parabolic PDEs, following their discretisation. For the most part finite element discretisations are considered, but other techniques are also discussed. Following discretisation of a PDE the two most frequently used nonlinear multigrid methods are Newton-Multigrid and the Full Approximation Scheme (FAS). These are both very efficient algorithms, and have the advantage that when they are applied to practical problems, their execution times scale linearly with the size of the problem being solved. Even though this has yet to be proved in theory for most problems, these methods have been widely adopted in practice in order to solve highly complex nonlinear (systems of) PDEs. Many research groups use either Newton-MG or FAS without much consideration as to which should be preferred, since both algorithms perform satisfactorily. In this thesis we address the question as to which method is likely to be more computationally efficient in practice. As part of this investigation the implementation of the algorithms is considered in a framework which allows the direct comparison of the computational effort of the two iterations. As well as this, the convergence properties of the methods are considered, applied to a variety of model problems. Extensive results are presented in the comparison, which are explained by available theory whenever possible. The strength and range of results presented allows us to confidently conclude that for a practical problem, discretised using a finite element discretisation, an improved efficiency and stability of a Newton-MG iteration, compared to an FAS iteration, is likely to be observed. The relative advantage of a Newton-MG method is likely to be larger the more complex the problem being solved becomes.

Item Type: Thesis (PhD)
Keywords: Multigrid, FAS, Newton-Multigrid, Nonlinear Multigrid, NMLM, Nonlinear Multilevel Method
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Identification Number/EthosID: uk.bl.ethos.643610
Depositing User: Mr K J Brabazon
Date Deposited: 02 Apr 2015 09:16
Last Modified: 25 Nov 2015 13:48
URI: http://etheses.whiterose.ac.uk/id/eprint/8481

You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page.

Actions (repository staff only: login required)