# On the Convergence of Adaptive Iterative Linearized Galerkin Methods

## TL;DR

This paper develops a unified convergence theory for adaptive iterative linearized Galerkin methods, encompassing various iterative schemes like Zarantonello, Kačanov, and Newton, applied to nonlinear equations in Hilbert spaces.

## Contribution

It introduces an abstract convergence framework for adaptive ILG schemes that unifies multiple iterative methods within a general discretization context.

## Key findings

- Convergence of the unified ILG approach is established theoretically.
- The framework is validated through numerical experiments on nonlinear conservation laws.
- Comparison of different linearization schemes demonstrates the effectiveness of the unified method.

## Abstract

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 that covers some prominent procedures (including the Zarantonello, Ka\v{c}anov 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 our previous work. 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.