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

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.