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Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai

Mathematical Sciences Spring Lecture Series

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's 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, …