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Mathematics of Computation

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Adaptive iterative linearization Galerkin methods for nonlinear problems

Authors: Pascal Heid and Thomas P. Wihler
Journal: Math. Comp. 89 (2020), 2707-2734
MSC (2010): Primary 47H10, 65N30; Secondary 47J25, 47H05, 49M15, 65J15
Published electronically: July 7, 2020
MathSciNet review: 4136544
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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

Thomas P. Wihler
Affiliation: Mathematics Institute, University of Bern, CH-3012 Switzerland
MR Author ID: 662940
ORCID: 0000-0003-1232-0637

Keywords: Numerical solution methods for nonlinear PDE, monotone problems, fixed-point iterations, linearization schemes, Kačanov method, Newton method, Galerkin discretizations, adaptive finite element methods, a posteriori error estimation
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
Article copyright: © Copyright 2020 American Mathematical Society