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Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient

Brabazon, Keeran J.; Hubbard, Matthew E.; Jimack, Peter K.

Authors

Keeran J. Brabazon

MATTHEW HUBBARD MATTHEW.HUBBARD@NOTTINGHAM.AC.UK
Professor of Computational and Applied Mathematics

Peter K. Jimack P.K.Jimack@leeds.ac.uk



Abstract

Nonlinear multigrid methods such as the Full Approximation Scheme (FAS) and Newton-multigrid (Newton-MG) are well established as fast solvers for nonlinear PDEs of elliptic and parabolic type. In this paper we consider Newton-MG and FAS iterations applied to second order differential operators with nonlinear diffusion coefficient. Under mild assumptions arising in practical applications, an approximation (shown to be sharp) of the execution time of the algorithms is derived, which demonstrates that Newton-MG can be expected to be a faster iteration than a standard FAS iteration for a finite element discretisation. Results are provided for elliptic and parabolic problems, demonstrating a faster execution time as well as greater stability of the Newton-MG iteration. Results are explained using current theory for the convergence of multigrid methods, giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice.

Journal Article Type Article
Publication Date Dec 31, 2014
Journal Computers and Mathematics with Applications
Print ISSN 0898-1221
Electronic ISSN 0898-1221
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 68
Issue 12A
APA6 Citation Brabazon, K. J., Hubbard, M. E., & Jimack, P. K. (2014). Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient. Computers and Mathematics with Applications, 68(12A), https://doi.org/10.1016/j.camwa.2014.11.002
DOI https://doi.org/10.1016/j.camwa.2014.11.002
Keywords Nonlinear multigrid; Newton’s method; Nonlinear diffusion
Publisher URL http://www.sciencedirect.com/science/article/pii/S0898122114005306
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0





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