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G. Gantner, A. Haberl, D. Praetorius, S. Schimanko:
"Rate optimal adaptive FEM with inexact solver for nonlinear operators";
Talk: WONAPDE 2019 - Sixth Chilean Workshop on Numerical Analysis of Partial Differential Equations, Concepción (invited); 01-21-2019 - 01-25-2019.

We aim to present our recent work [G. Gantner, A. Haberl, D. Praetorius, B. Stiftner. Rate optimal adaptive FEM with inexact solver for nonlinear operators. IMA Journal of Numerical Analysis, 38, 1797-1831, 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 [S. Congreve and T.P. Wihler. Iterative Galerkin discretizations for strongly monotone problems. Journal of Computational and Applied Mathematics, 311, 457-472, 2017]. Unlike prior works [E.M. Garau, P. Morin, and C. Zuppa. Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type. Numerical Mathematics: Theory, Methods and Applications, 5(2), 131-156, 2012], 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 of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments con firm the theoretical results.