Energy contraction and optimal convergence of adaptive iterative linearized finite element methods

TL;DR

This paper demonstrates that adaptive iterative linearized finite element methods (AILFEMs) achieve full linear convergence with optimal algebraic rates for solving nonlinear equations in Hilbert spaces, based on an energy contraction property.

Contribution

It establishes the energy contraction property for AILFEMs in strongly monotone problems, leading to optimal convergence rates and computational efficiency.

Findings

AILFEMs exhibit full linear convergence.

Optimal algebraic rates are achieved with respect to degrees of freedom.

Convergence is proven under an energy contraction framework.

Abstract

We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [Heid & Wihler, Math. Comp. 89 (2020), Calcolo 57 (2020)] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [Gantner et al., arXiv:2003.10785]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.