Energy Contraction and Optimal Convergence of Adaptive Iterative Linearized Finite Element Methods

We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 (2020), no. 326, 2707-2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 (2020), Paper No. 24] 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 [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the over-all computational costs, preprint (2020)]. 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.

Automated summary

Through previous research, it has been found that the methodology proposed by Heid and Wihler exhibits an energy contraction property in the context of strongly monotone problems, which is crucial for achieving linear convergence. Specifically, adaptive iterative linearized finite element methods lead to full linear convergence with optimal algebraic rates in terms of degrees of freedom and total computational time.