A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.

中文翻译：

---

非线性问题的自适应迭代线性化伽辽金方法

存在多种用于求解非线性方程（在希尔伯特空间中）的（定点）迭代方法。在许多情况下，这种方案可以被解释为迭代局部线性化方法，正如将显示的，可以通过将合适的预处理算子应用于原始（非线性）方程来获得。基于这一观察，我们将推导出一个统一的抽象框架，该框架恢复了一些突出的迭代方案。特别是，对于 Lipschitz 连续和强单调算子，我们推导出一般收敛分析。此外，在非线性偏微分方程的数值求解方案的背景下，我们提出了迭代线性化方法和经典伽辽金离散化方法的组合，从而产生了所谓的迭代线性化伽辽金 (ILG) 方法。此外，仍然在抽象层面上，基于两种不同的椭圆重建技术，我们推导出后验误差估计，分别考虑离散化和线性化误差。此外，我们提出了一种自适应算法，它提供了这两种效果之间的有效相互作用。此外，ILG 方法将应用于拟线性椭圆方程的有限元离散化的特定上下文，并进行一些数值实验。我们提出了一种自适应算法，它提供了这两种效果之间的有效相互作用。此外，ILG 方法将应用于拟线性椭圆方程的有限元离散化的特定上下文，并进行一些数值实验。我们提出了一种自适应算法，它提供了这两种效果之间的有效相互作用。此外，ILG 方法将应用于拟线性椭圆方程的有限元离散化的特定上下文，并进行一些数值实验。