Document Type
Video
Publication Date
4-7-2021
Keywords
Root finding, motivating applications, nonlinear, Newton methods, fluid problems, solid problems, biomechanics
Abstract
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, 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.
Citation
Cai, X. (2021). Lecture 07: Nonlinear Preconditioning Methods and Applications. Mathematical Sciences Spring Lecture Series. Retrieved from https://scholarworks.uark.edu/mascsls/12
Included in
Dynamic Systems Commons, Non-linear Dynamics Commons, Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons
Comments
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