NDLI: Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kheriji, W. Gander, M. J. Dolean, V. Masson, R. Kwok, F.
Copyright Year	2016
Abstract	For linear problems, domain decomposition methods can be used directly as iterative solvers but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration and thus converges much faster. We show in this paper that also for nonlinear problems, domain decomposition methods can be used either directly as iterative solvers or as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (restricted additive Schwarz preconditioned exact Newton), which is similar to ASPIN (additive Schwarz preconditioned inexact Newton) but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two-level nonlinear iterative domain decomposition method and a two level RASPEN nonlinear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a nonlinear diffusion problem.
Sponsorship	Research Grants Council, University Grants Committee. National Natural Science Foundation of China
Starting Page	A3357
Ending Page	A3380
Page Count	24
File Format	PDF
ISSN	10648275
DOI	10.1137/15M102887X
e-ISSN	10957197
Issue Number	6
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-11-01
Access Restriction	Subscribed
Subject Keyword	Solution of discretized equations Iterative methods for linear systems two-level nonlinear Schwarz methods nonlinear preconditioning preconditioning Newton's method Multigrid methods; domain decomposition
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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