Unconditional full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs

TL;DR

This paper introduces an adaptive linearization finite element method for nonlinear PDEs that guarantees full linear convergence and optimal complexity, regardless of adaptivity parameter choices, through a novel parameter-free stopping criterion.

Contribution

It develops a parameter-robust adaptive FEM algorithm with guaranteed full linear convergence and optimal complexity for nonlinear PDEs, using a new stopping criterion.

Findings

Guarantees full R-linear convergence for any adaptivity parameter.

Ensures optimal computational complexity with small adaptivity parameters.

Introduces a novel parameter-free algebraic stopping criterion.

Abstract

We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Ka\v{c}anov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, parameter-robust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.