We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 2020, 326, 2707–2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 2020, Paper No. 24] 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 [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the overall computational costs, preprint 2020]. 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.

中文翻译：

---

自适应迭代线性化有限元方法的能量收缩和最优收敛

我们重新审视Hilbert空间中非线性方程组迭代求解的统一方法。我们的主要观察结果是[P. Heid和T.P. Wihler，针对非线性问题的自适应迭代线性化Galerkin方法，数学。比较 89 2020，326，2707–2734; P. Heid和T. P. Wihler，在自适应迭代线性化Galerkin方法的收敛性上，《 Calcolo 57 2020，第24号论文》满足（抽象）强单调问题的能量收缩特性。反过来，此属性是[G中最近的收敛分析中的关键要素。Gantner，A。Haberl，D。Praetorius和S. Schimanko，关于整体计算成本的自适应有限元方法的速率最优性，预印本2020年]。尤其是，