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

On full linear convergence and optimal complexity of adaptive FEM with inexact solver