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M. Brunner, G. Gantner, M. Innerberger, D. Praetorius:
"Adaptive FEM with quasi-optimal cost for nonlinear PDEs";
Keynote Lecture: GATIPOR Workshop 2022 on Interplay of discretization and algebraic solvers: a posteriori error estimates and adaptivity, Paris (invited); 06-08-2022 - 06-10-2022.

We consider nonlinear elliptic PDEs with strongly montone nonlinearity. We apply an adaptive
finite element method, which steers the linearization as well as the iterative solution of the
arising linear finite element systems. We prove that the proposed algorithm guarantees full
linear convergence (i.e., linear convergence in each step, independently of the algorithmic
decision for mesh-refinement, linearization, or algebraic solver step). For sufficiently small
adaptivity parameters, this allows to guarantee optimal convergence with respect to the overall
computational work (i.e., the computational time).

The talk is based on joint work [1, 2, 3].

[1] G. Gantner, A. Haberl, D. Praetorius, S. Schimanko: Rate optimality of adaptive finite element
methods with respect to the overall computational costs, Mathematics of Computation, 90
(2021), 2011-2040.

[2] A. Haberl, D. Praetorius, S. Schimanko, M. Vohralik: Convergence and quasi-optimal cost of
adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver,
Numerische Mathematik, 147 (2021), 679-725.

[3] P. Heid, D. Praetorius, T. Wihler: Energy contraction and optimal convergence of adaptive
iterative linearized finite element methods, Computational Methods in Applied Mathematics, 21
(2021), 407-422.