We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton\u27s method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, we show that fast convergence can be restored. In this talk we first review the basic algorithms, and then discuss some recent progress in the applications of nonlinear preconditioners for some difficult problems arising in computational mechanics including both fluid dynamics and solid mechanics
