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：任何偏微分方程数值方案的最终目标，都是在准最小计算时间内计算出用户指定精度的近似解。为此，在算法层面，标准自适应有限元法将非精确求解器与嵌套迭代相结合，并采用能够平衡不同误差分量的判别性停止准则。保证AFEM关于整体计算成本具有最优收敛阶的分析，关键在于一个合适的拟误差量具有R-线性收敛性的概念。本文通过引入新的证明策略，解决了以往方法的若干不足。首先，为使基础分析成立，算法需要多个精细调谐的参数。通过重新设计标准论证思路并引入R-线性收敛的可和性准则，我们得以消除对这些参数的限制。其次，用[Feischl, Math. Comp., 91 (2022)]中提出的广义拟正交性概念，取代了通常的（拟）勾股定理假设。重要的是，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。数值实验研究了适应性参数的选择。