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Convergence of goal-oriented adaptive finite element methods for semilinear problems

Holst, Michael ; Pollock, Sara ; Zhu, Yunrong

Computing and visualization in science, 2015-02, Vol.17 (1), p.43-63 [Periódico revisado por pares]

Berlin/Heidelberg: Springer Berlin Heidelberg

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  • Título:
    Convergence of goal-oriented adaptive finite element methods for semilinear problems
  • Autor: Holst, Michael ; Pollock, Sara ; Zhu, Yunrong
  • Assuntos: Algorithms ; Calculus of Variations and Optimal Control; Optimization ; Computational Mathematics and Numerical Analysis ; Computer Applications in Chemistry ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Visualization
  • É parte de: Computing and visualization in science, 2015-02, Vol.17 (1), p.43-63
  • Descrição: In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory.
  • Editor: Berlin/Heidelberg: Springer Berlin Heidelberg
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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