A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work [16] that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in [16]. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws.

中文翻译：

---

关于自适应迭代线性化Galerkin方法的收敛性

存在多种用于求解非线性方程的不同（定点）迭代方法。在这项工作中，我们将从先前的工作[16]中重新审视希尔伯特空间中的统一迭代方案，该方案涵盖了一些著名的过程（包括Zarantonello，Kačanov和Newton迭代方法）。结合适当的离散化方法，所谓的（自适应）迭代线性化Galerkin（ILG）方案获得。本文的主要目的是推导[16]中提出的统一ILG方法（基于通用自适应Galerkin离散化方法）的抽象收敛理论。在强单调平稳守恒律的自适应有限元离散化的背景下，将对上述三种迭代线性化方案的理论结果进行测试和比较。