SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1404-B1428, January 2020.
We develop a multilayer nonlinear elimination preconditioned inexact Newton method for a nonlinear algebraic system of equations, and a target application is the three-dimensional steady-state incompressible Navier--Stokes equations at high Reynolds numbers. Nonlinear steady-state problems are often more difficult to solve than time-dependent problems because the Jacobian matrix is less diagonally dominant, and a good initial guess from the previous time step is not available. For such problems, Newton-like methods may suffer from slow convergence or stagnation even with globalization techniques such as line search. In this paper, we introduce a cascadic multilayer nonlinear elimination approach based on feedback from intermediate solutions to improve the convergence of Newton iteration. Numerical experiments show that the proposed algorithm is superior to the classical inexact Newton method and other single layer nonlinear elimination approaches in terms of the robustness and efficiency. Using the proposed nonlinear preconditioner with a highly parallel domain decomposition framework, we demonstrate that steady solutions of the Navier--Stokes equations with Reynolds numbers as large as 7,500 can be obtained for the lid-driven cavity flow problem in three dimensions without the use of any continuation methods.

中文翻译：

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三维稳态稳态不可压缩流动问题的多层非线性消除预处理不精确牛顿法

SIAM科学计算杂志，第42卷，第6期，第B1404-B1428页，2020年1月。
我们针对非线性代数方程组开发了多层非线性消除预处理不精确牛顿法，其目标应用是高雷诺数下的三维稳态不可压缩Navier-Stokes方程。非线性稳态问题通常比时间相关的问题更难解决，因为雅可比矩阵的对角线支配性较小，并且无法获得上一个时间步长的良好初始猜测。对于此类问题，即使使用诸如行搜索之类的全球化技术，类牛顿法也可能会出现收敛缓慢或停滞的情况。在本文中，我们引入了基于中间解的反馈的级联多层非线性消除方法，以提高牛顿迭代的收敛性。数值实验表明，该算法在鲁棒性和效率上均优于经典的非精确牛顿法和其他单层非线性消除方法。使用具有高度并行域分解框架的拟议非线性预处理器，我们证明了对于三维驱动的盖驱动腔流动问题，无需使用Navier-Stokes方程，该方程的雷诺数高达7,500即可得到稳态解。任何延续方法。