## Adaptive iterative linearization Galerkin methods for nonlinear problems

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Pascal Heid and Thomas P. Wihler
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## Abstract:

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative*local linearization*methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called

*iterative linearization Galerkin (ILG)*methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.

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## Additional Information

**Pascal Heid**- Affiliation: Mathematics Institute, University of Bern, CH-3012 Switzerland
- ORCID: 0000-0003-4227-4053
- Email: pascal.heid@math.unibe.ch
**Thomas P. Wihler**- Affiliation: Mathematics Institute, University of Bern, CH-3012 Switzerland
- MR Author ID: 662940
- ORCID: 0000-0003-1232-0637
- Email: wihler@math.unibe.ch
- Received by editor(s): August 15, 2018
- Received by editor(s) in revised form: October 15, 2019
- Published electronically: July 7, 2020
- Additional Notes: The authors acknowledge the financial support of the Swiss National Science Foundation under grant no. 200021_182524
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**89**(2020), 2707-2734 - MSC (2010): Primary 47H10, 65N30; Secondary 47J25, 47H05, 49M15, 65J15
- DOI: https://doi.org/10.1090/mcom/3545
- MathSciNet review: 4136544