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满足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层面集成了不精确求解器与嵌套迭代，并通过区分不同误差分量的终止准则实现平衡。分析确保自适应有限元法在整体计算代价下的最优收敛阶，关键依赖于适当准误差量的R-线性收敛性概念。本研究通过引入新证明策略，弥补了先前方法的若干不足。首先，原算法需要多个精细调节参数以保证底层分析的有效性。通过重新设计标准推理框架并引入R-线性收敛的可和性准则，我们得以解除对这些参数的限制。其次，将通常所需的（准）毕达哥拉斯恒等式替换为[Feischl, Math. Comp., 91 (2022)]中提出的广义准正交性概念。重要的是，这为将分析推广至能量最小化框架之外的一般inf-sup稳定问题铺平了道路。数值实验探讨了自适应参数的选择策略。