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G. Gantner, D. Praetorius, S. Schimanko:
"Rate optimal adaptive FEM with inexact solver for nonlinear operators";
Keynote Lecture: BI.discrete - Numerical Analysis in Bielefeld, Bielefeld (invited); 2019-09-25 - 2019-09-26.

We present our recent work [Gantner et al., IMA J Numer Anal 38, 2018], where we prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. We consider an algorithm proposed by [Congreve et al., J Comp Appl Math 311, 2017]. Unlike prior works, e.g., [Carstensen et al., Comp Math Appl 67, 2014], our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of Picard iterations is uniformly bounded in generic cases. Finally, we aim to discuss that the overall computational cost is optimal. Numerical experiments confirm the theoretical results.