The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. First, the algorithm requires several fine-tuned parameters in order to make the underlying analysis work. A redesign of the standard line of reasoning and the introduction of a summability criterion for R-linear convergence allows us to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91 (2022)]. Importantly, this paves the way towards extending the analysis to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.

翻译：偏微分方程数值方案的最终目标是以准最小计算时间计算用户指定精度的近似解。为此，标准自适应有限元方法在算法上集成了不精确求解器和嵌套迭代，并通过区分性停止准则平衡不同误差分量。确保自适应有限元方法在整体计算成本下保持最优收敛阶的分析，关键依赖于适当准误差量的R-线性收敛概念。本文通过引入新的证明策略，克服了先前方法的若干缺陷。首先，该算法需要多个精细调谐的参数才能使底层分析成立。通过对标准推理路线的重新设计，并引入R-线性收敛的可和性准则，我们得以取消对这些参数的限制。其次，常见的（拟）勾股恒等式假设被[Feischl, Math. Comp., 91 (2022)]中提出的广义拟正交性概念所取代。重要的是，这为将分析推广到超出能量极小化设定的一般inf-sup稳定问题铺平了道路。数值实验研究了自适应参数的选择。