Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
Journal of Computational Physics
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
An Analysis of Approximate Nonlinear Elimination
SIAM Journal on Scientific Computing
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Convergence Analysis of Pseudo-Transient Continuation
SIAM Journal on Numerical Analysis
Parallel Newton--Krylov--Schwarz Algorithms for the Transonic Full Potential Equation
SIAM Journal on Scientific Computing
NITSOL: A Newton Iterative Solver for Nonlinear Systems
SIAM Journal on Scientific Computing
A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems
SIAM Journal on Scientific Computing
Nonlinearly Preconditioned Inexact Newton Algorithms
SIAM Journal on Scientific Computing
Globalized Newton-Krylov-Schwarz Algorithms and Software for Parallel Implicit CFD
International Journal of High Performance Computing Applications
Journal of Computational Physics
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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 Navier-Stokes equations with high Reynolds numbers and on machines with large numbers of processors.