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";

in: "ASC Report 12/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 40.

We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach-Picard linearization, and a contractive linear algebraic solver. We in particular identify topping

criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) Banach-Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach-Picard iteration that leave an amount of linearization error that is not harmful for the residual a-posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after

some amount of steps of mesh-refinement/linerization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive

algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments

support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.

elliptic boundary value problem, monotone nonlinearity, strong monotonicity, finite element method, Banach-Picard linearization, algebraic resolution, inexact solver, stopping criterion, a posteriori error estimate, adaptive mesh-refinement, contraction,

http://www.asc.tuwien.ac.at/preprint/2020/asc12x2020.pdf

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