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Adaptive iterative linearization Galerkin methods for nonlinear problems.

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Title: Adaptive iterative linearization Galerkin methods for nonlinear problems.
Authors: Heid, Pascal1 (AUTHOR), Wihler, Thomas P.1 (AUTHOR)
Source: Mathematics of Computation. Nov2020, Vol. 89 Issue 326, p2707-2734. 28p.
Subjects: Galerkin methods, Nonlinear equations, Nonlinear differential equations, Partial differential equations, Quasilinearization, Algorithms, Monotone operators
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. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 145198878
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  Data: Adaptive iterative linearization Galerkin methods for nonlinear problems.
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  Data: <searchLink fieldCode="AR" term="%22Heid%2C+Pascal%22">Heid, Pascal</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Wihler%2C+Thomas+P%2E%22">Wihler, Thomas P.</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Nov2020, Vol. 89 Issue 326, p2707-2734. 28p.
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  Data: <searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+equations%22">Nonlinear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+differential+equations%22">Nonlinear differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Quasilinearization%22">Quasilinearization</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Monotone+operators%22">Monotone operators</searchLink>
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  Data: 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1090/mcom/3545
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      – Code: eng
        Text: English
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        PageCount: 28
        StartPage: 2707
    Subjects:
      – SubjectFull: Galerkin methods
        Type: general
      – SubjectFull: Nonlinear equations
        Type: general
      – SubjectFull: Nonlinear differential equations
        Type: general
      – SubjectFull: Partial differential equations
        Type: general
      – SubjectFull: Quasilinearization
        Type: general
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Monotone operators
        Type: general
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      – TitleFull: Adaptive iterative linearization Galerkin methods for nonlinear problems.
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            – D: 01
              M: 11
              Text: Nov2020
              Type: published
              Y: 2020
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              Value: 326
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            – TitleFull: Mathematics of Computation
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