Quasi-optimal complexity of iterative Galerkin methods driven by an elliptic reconstruction error estimator
For researchers in numerical analysis and scientific computing, this work provides a rigorous theoretical foundation for adaptive iterative solvers for quasilinear elliptic PDEs, establishing optimal complexity guarantees.
This paper presents the first comprehensive convergence analysis of an iterative Galerkin method for quasilinear elliptic problems, using an elliptic reconstruction error estimator to drive adaptive mesh refinement. The method achieves unconditional full R-linear convergence and, for small adaptivity parameters, optimal convergence rates with respect to degrees of freedom and quasi-optimal complexity.
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.