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

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

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.

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

• Journal Article Type: Article
• Acceptance Date: Nov 4, 2014
• Online Publication Date: Nov 22, 2014
• Publication Date: Dec 31, 2014
• Deposit Date: Feb 27, 2017
• Publicly Available Date: Feb 27, 2017
• Journal: Computers and Mathematics with Applications
• Print ISSN: 0898-1221
• Electronic ISSN: 0898-1221
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 68
• Issue: 12A
• DOI: https://doi.org/10.1016/j.camwa.2014.11.002
• Keywords: Nonlinear multigrid; Newton’s method; Nonlinear diffusion
• Public URL: https://nottingham-repository.worktribe.com/output/740338
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0898122114005306