Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient
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...
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| Format: | Article |
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Elsevier
2014
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| Online Access: | https://eprints.nottingham.ac.uk/40817/ |
| _version_ | 1848796139847942144 |
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| author | Brabazon, Keeran J. Hubbard, Matthew E. Jimack, Peter K. |
| author_facet | Brabazon, Keeran J. Hubbard, Matthew E. Jimack, Peter K. |
| author_sort | Brabazon, Keeran J. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:43:14Z |
| format | Article |
| id | nottingham-40817 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:43:14Z |
| publishDate | 2014 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-408172020-05-04T16:58:13Z https://eprints.nottingham.ac.uk/40817/ Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient Brabazon, Keeran J. Hubbard, Matthew E. Jimack, Peter K. 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. Elsevier 2014-12-31 Article PeerReviewed Brabazon, Keeran J., Hubbard, Matthew E. and Jimack, Peter K. (2014) Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient. Computers and Mathematics with Applications, 68 (12A). pp. 1619-1634. ISSN 0898-1221 Nonlinear multigrid; Newton’s method; Nonlinear diffusion http://www.sciencedirect.com/science/article/pii/S0898122114005306 doi:10.1016/j.camwa.2014.11.002 doi:10.1016/j.camwa.2014.11.002 |
| spellingShingle | Nonlinear multigrid; Newton’s method; Nonlinear diffusion Brabazon, Keeran J. Hubbard, Matthew E. Jimack, Peter K. Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title | Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title_full | Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title_fullStr | Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title_full_unstemmed | Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title_short | Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| title_sort | nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient |
| topic | Nonlinear multigrid; Newton’s method; Nonlinear diffusion |
| url | https://eprints.nottingham.ac.uk/40817/ https://eprints.nottingham.ac.uk/40817/ https://eprints.nottingham.ac.uk/40817/ |