Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity

TL;DR

This paper proves that an adaptive finite element method combined with an optimally preconditioned GMRES solver achieves optimal computational complexity and guaranteed convergence for second-order elliptic PDEs.

Contribution

It introduces a new adaptive algorithm that jointly refines the mesh and controls the GMRES solver, ensuring optimal complexity and unconditional convergence.

Findings

Algorithm guarantees unconditional convergence regardless of parameters.

Quasi-error decays at optimal rates relative to computational complexity.

Adaptive feedback-control monitors and ensures full R-linear convergence.

Abstract

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.