Quasi-optimal complexity of iterative Galerkin methods driven by an elliptic reconstruction error estimator

TL;DR

This paper presents a comprehensive convergence analysis of an adaptive iterative Galerkin method for quasilinear elliptic problems, achieving optimal convergence rates and complexity driven by an elliptic reconstruction error estimator.

Contribution

It introduces the first detailed convergence analysis for this adaptive Galerkin method with elliptic reconstruction error-driven mesh refinement.

Findings

Proves unconditional full R-linear convergence of the combined linearization and discretization errors.

Establishes optimal convergence rates with respect to degrees of freedom.

Demonstrates quasi-optimal complexity matching the best possible computational cost.

Abstract

We study an iterative Galerkin method for quasilinear elliptic problems in the Browder-Minty setting. The resulting discrete nonlinear systems are solved by linearization via a (damped) Zarantonello iteration. Unlike prior work, adaptive mesh refinement is driven by an elliptic reconstruction error estimator, which is natural in the sense that the a posteriori bounds for the linearization and discretization errors are well separated. For this setting, we present the first comprehensive convergence analysis of the corresponding algorithm. We prove unconditional full R-linear convergence of a suitable quasi-error that combines linearization and discretization errors. For sufficiently small adaptivity parameters, we further establish optimal convergence rates with respect to the number of degrees of freedom and quasi-optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost. Numerical experiments underpin the theoretical findings.