NDLI: Inexact Newton Methods with Restricted Additive Schwarz Based Nonlinear Elimination for Problems with High Local Nonlinearity

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Li, Xuefeng Cai, Xiao-Chuan
Copyright Year	2011
Abstract	The classical inexact Newton algorithm is an efficient and popular technique for solving large sparse nonlinear systems of equations. When the nonlinearities in the system are well balanced, a near quadratic convergence is often observed; however, if some of the equations are much more nonlinear than the others in the system, the convergence is much slower. The slow convergence (or sometimes divergence) is often determined by the small number of equations in the system with the highest nonlinearities. The idea of nonlinear preconditioning has been proven to be very useful. Through subspace nonlinear solves, the local high nonlinearities are removed, and the fast convergence can then be restored when the inexact Newton algorithm is called after the preconditioning. Recently a left preconditioned inexact Newton method was proposed in which the nonlinear function is replaced by a preconditioned function with more balanced nonlinearities. In this paper, we combine an inexact Newton method with a restricted additive Schwarz based nonlinear elimination. The new approach is easier to implement than the left preconditioned method since the nonlinear function does not have to be replaced, and, furthermore, the nonlinear elimination step does not have to be called at every outer Newton iteration. We show numerically that it performs well for, as an example, solving the incompressible NavierStokes equations with high Reynolds numbers and on machines with large numbers of processors.
Starting Page	746
Ending Page	762
Page Count	17
File Format	PDF
ISSN	10648275
DOI	10.1137/080736272
e-ISSN	10957197
Issue Number	2
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-03-17
Access Restriction	Subscribed
Subject Keyword	domain decomposition nonlinear restricted additive Schwarz incompressible flows Global methods, including homotopy approaches Stability and convergence of numerical methods nonlinear equations Systems of equations Solution of discretized equations Iterative methods for linear systems nonlinear elimination inexact Newton method Parallel computation Multigrid methods; domain decomposition parallel computing Finite difference methods Navier-Stokes equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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