Adaptive iterative linearization Galerkin methods for nonlinear problems

Heid, Pascal; Wihler, Thomas P. (2020). Adaptive iterative linearization Galerkin methods for nonlinear problems. Mathematics of computation, 89(326), pp. 2707-2734. American Mathematical Society 10.1090/mcom/3545

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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.

Item Type:

Journal Article (Original Article)


08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Heid, Pascal, Wihler, Thomas


500 Science > 510 Mathematics




American Mathematical Society


[4] Swiss National Science Foundation




Sebastiano Don

Date Deposited:

10 Feb 2021 15:10

Last Modified:

05 Dec 2022 15:45

Publisher DOI:


ArXiv ID:


Uncontrolled Keywords:

Numerical solution methods for nonlinear PDE, monotone problems, fixed-point iterations, linearization schemes, Kaˇcanov method, Newton method, Galerkin discretizations, adaptive finite element methods, a posteriori error estimation




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