SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2526-A2554, January 2021.
In many multiphysics applications, an ultimate quantity of interest can be written as a linear functional of the solution to the discretized governing nonlinear partial differential equations and finding a sufficiently accurate pointwise solution may be regarded as a step toward that end. In this paper, we derive a posteriori approximate error bounds for linear functionals corresponding to quantities of interest using two kinds of nonlinear preconditioning techniques. Nonlinear preconditioning, such as the inexact Newton with backtracking and nonlinear elimination algorithm and the multiplicative Schwarz preconditioned inexact Newton algorithm, may be effective in improving global convergence for Newton's method. It may prevent stagnation of the nonlinear residual norm and reduce the number of solutions of large ill-conditioned linear systems involving a global Jacobian required at each nonlinear iteration. We illustrate the effectiveness of the new bounds using canonical nonlinear PDE models: a flame sheet model and a nonlinear coupled lid-driven cavity problem.

中文翻译：

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非线性预处理偏微分方程解的近似误差界限

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2526-A2554 页，2021 年 1 月。
在许多多物理场应用中，最终感兴趣的量可以写成离散化控制非线性偏微分方程解的线性函数，找到足够准确的逐点解可以被视为实现这一目标的一步。在本文中，我们使用两种非线性预处理技术推导出与感兴趣量相对应的线性函数的后验近似误差界限。非线性预处理，例如带有回溯和非线性消除算法的不精确牛顿算法和乘法 Schwarz 预处理不精确牛顿算法，可以有效地提高牛顿方法的全局收敛性。它可以防止非线性残差范数的停滞，并减少涉及每次非线性迭代所需的全局雅可比行列式的大型病态线性系统的解数。我们使用规范非线性 PDE 模型来说明新边界的有效性：火焰板模型和非线性耦合盖驱动腔问题。