Nonlinearly Preconditioned Inexact Newton Algorithms

Inexact Newton algorithms are commonly used for solving large sparse nonlinear system of equations F(u^)=0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of $\|F\|$, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u^, one may want to solve instead an equivalent nonlinearly preconditioned system F(u^)=0 whose nonlinearities are more balanced. In this paper, we propose and study a nonlinear additive Schwarz-based parallel nonlinear preconditioner and show numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails.