Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity
Provides a rigorous theoretical guarantee of optimal complexity for adaptive FEM with iterative solvers, addressing a key bottleneck in computational PDEs.
This paper proves that an adaptive finite element method combined with an optimally preconditioned GMRES solver achieves optimal computational complexity for solving general second-order linear elliptic PDEs. The algorithm guarantees unconditional convergence and optimal convergence rates when adaptivity parameters are chosen sufficiently small.
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.