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

TL;DR

This paper establishes full linear convergence and optimal complexity for adaptive finite element methods with inexact solvers, introducing a new proof strategy that relaxes previous assumptions and extends applicability.

Contribution

The work presents a novel proof approach for AFEM convergence, removing parameter restrictions and generalizing to broader problem classes beyond energy minimization.

Findings

Proves full linear convergence of AFEM with inexact solvers.

Achieves optimal complexity bounds for adaptive FEM.

Extends analysis to general inf-sup stable problems.

Abstract

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