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A NUMERICAL ENERGY REDUCTION APPROACH FOR SEMILINEAR DIFFUSION-REACTION BOUNDARY VALUE PROBLEMS BASED ON STEADY-STATE ITERATIONS.

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Title: A NUMERICAL ENERGY REDUCTION APPROACH FOR SEMILINEAR DIFFUSION-REACTION BOUNDARY VALUE PROBLEMS BASED ON STEADY-STATE ITERATIONS.
Authors: AMREIN, MARIO1 mario.amrein@zhaw.ch, HEID, PASCAL2 pascal.heid@maths.ox.ac.uk, WIHLER, THOMAS P.3 wihler@math.unibe.ch
Source: SIAM Journal on Numerical Analysis. 2023, Vol. 61 Issue 2, p755-783. 29p.
Subjects: Boundary value problems, Mathematical sequences, Numerical analysis, Finite element method
Abstract: We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilized) linearized discrete problems, and tends to a critical point of the underlying energy functional in a stable way. Simultaneously, the finite-dimensional approximation spaces are adaptively refined. This is implemented in terms of a new mesh refinement strategy in the context of finite element discretizations, which again relies on the energy structure of the problem under consideration. In particular, in contrast to more traditional approaches, it does not involve any a posteriori error estimators, and is based on local energy reduction indicators instead. In combination, the resulting adaptive algorithm consists of an iterative linearization procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge toward a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Numerical Analysis is the property of Society for Industrial & Applied Mathematics 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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  Data: We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilized) linearized discrete problems, and tends to a critical point of the underlying energy functional in a stable way. Simultaneously, the finite-dimensional approximation spaces are adaptively refined. This is implemented in terms of a new mesh refinement strategy in the context of finite element discretizations, which again relies on the energy structure of the problem under consideration. In particular, in contrast to more traditional approaches, it does not involve any a posteriori error estimators, and is based on local energy reduction indicators instead. In combination, the resulting adaptive algorithm consists of an iterative linearization procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge toward a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Numerical Analysis is the property of Society for Industrial & Applied Mathematics 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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        Value: 10.1137/22M1478586
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        Text: English
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        PageCount: 29
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      – SubjectFull: Boundary value problems
        Type: general
      – SubjectFull: Mathematical sequences
        Type: general
      – SubjectFull: Numerical analysis
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      – SubjectFull: Finite element method
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      – TitleFull: A NUMERICAL ENERGY REDUCTION APPROACH FOR SEMILINEAR DIFFUSION-REACTION BOUNDARY VALUE PROBLEMS BASED ON STEADY-STATE ITERATIONS.
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              M: 03
              Text: 2023
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