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：偏微分方程数值计算方案的最终目标是在准最小计算时间内获得用户指定精度的近似解。为实现这一目标，标准自适应有限元法在算法层面集成了非精确求解器与嵌套迭代策略，并通过精密的停止准则平衡不同误差分量。为确保AFEM相对于总计算成本具有最优收敛阶的分析，其关键依赖于一个适当准误差量的R-线性收敛概念。本研究通过引入新的证明策略，解决了以往方法的若干缺陷。首先，现有算法需要多个精细调谐的参数才能使基础分析成立。通过重新设计标准论证路径并引入R-线性收敛的可和性准则，我们消除了对这些参数的限制。其次，用[Feischl, Math. Comp., 91 (2022)]中提出的广义准正交概念替代了传统的（准）勾股恒等式假设。这为将分析扩展到能量最小化框架之外的一般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策略。