We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [Heid & Wihler, Math. Comp. 89 (2020), Calcolo 57 (2020)] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [Gantner et al., arXiv:2003.10785]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.

翻译：我们重新审视了希尔伯特空间非线性方程式迭代解决方案的统一方法。我们的主要看法是,[海德和维赫勒, Math.comp. 89 (2020), Calcolo 57 (2020)] 的一般方法在(抽象)强烈的单质问题中满足了一种能量收缩财产。而这种财产则是最近在[甘特纳等人,arXiv:2003.10785] 进行的趋同分析中的关键成分。我们特别推断,适应性迭代线性线性有限要素方法(AILFEMs)在自由度和总计算时间方面,与最佳代数率完全线性趋同。