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：对于偏微分方程（PDE）的任何数值方案，其最终目标是在准最小计算时间内，计算满足用户指定精度要求的近似解。为此，算法上，标准自适应有限元方法（AFEM）集成了不精确求解器和嵌套迭代，并采用区分误差分量的判断性停止准则。确保AFEM在整体计算成本方面具有最优收敛阶的分析，关键依赖于适当拟误差量的R-线性收敛概念。本文通过引入新的证明策略，解决了先前方法的若干不足。首先，原算法需要多个精细调节的参数以使底层分析可行。通过重新设计标准推理路线并引入R-线性收敛的可和性准则，我们得以去除对这些参数的限制。其次，我们将通常的（拟）勾股恒等式假设替换为[Feischl, Math. Comp., 91 (2022)]中提出的广义拟正交性概念。重要的是，这为将分析推广至能量极小化框架之外的一般inf-sup稳定问题铺平了道路。数值实验探究了自适应参数的选择。