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

2026/05/31 by Maximilian Brunner, Gregor Gantner, Christoph Lietz +1
Computer Science · Mathematics · #cs.NA #math.NA #msc:41A25 #msc:65N15 #msc:65N30 #msc:65N50 #msc:65Y20

paper · pdf · doi:10.48550/arxiv.2605.20057

arxiv created 2026/08/06 · arxiv updated 2026/08/07

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.

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