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            <funder>This research was funded by the Austrian Science Fund (FWF) projects 10.55776/F65 (SFB F65 “Taming complexity in PDE systems”), 10.55776/I6802 (international project “Functional error estimates for PDEs on unbounded domains”), 10.55776/PAT3446525 (standalone project “Adaptive Uzawa-type FEM for nonlinear PDEs”), and 10.55776/PAT3699424 (standalone project “Optimal robust solvers for reliable and efficient AFEMs”) and by ANID FONDECYT Regular 1250070 project.</funder>
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                <term xml:lang="en">Adaptive finite element method</term>
                <term xml:lang="en">Non-symmetric problem</term>
                <term xml:lang="en">Optimal convergence rates</term>
                <term xml:lang="en">Cost-optimality</term>
                <term xml:lang="en">Inexact solver</term>
                <term xml:lang="en">Full linear convergence</term>
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              <p>We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers the local mesh-refinement as well as the termination of a generalized minimal residual solver (GMRES) with optimal preconditioner to solve the arising non-symmetric finite element systems. Algorithmic interplay of mesh-refinement and iterative solver is shown to be optimal: A natural and fully computable quasi-error monitoring discretization error and algebraic solver error guarantees unconditional convergence for any choice of adaptivity parameters, i.e., the algorithm cannot fail to converge. This is ensured algorithmically via a novel adaptive feedback-control for the solver-termination parameter that monitors and ensures full R-linear convergence. Finally, the quasi-error even decays with optimal rates with respect to the overall computational complexity if the adaptivity parameters are chosen sufficiently small.</p>
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