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 Schwarzbased 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.