Pré-Publication, Document De Travail Année : 2026

Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity

Résumé

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.

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hal-05596791 , version 1 (20-04-2026)

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  • HAL Id : hal-05596791 , version 1

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Thomas Führer, Paula Hilbert, Ani Miraçi, Dirk Praetorius. Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity. 2026. ⟨hal-05596791⟩
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