计算复杂度最优的目标导向自适应有限元方法

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2023-01-01 Epub Date: 2022-11-16 DOI:10.1007/s00211-022-01334-8
Roland Becker, Gregor Gantner, Michael Innerberger, Dirk Praetorius
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引用次数: 0

摘要

我们考虑了线性对称椭圆 PDE 和线性目标函数。我们设计并分析了一种以目标为导向的自适应有限元方法,该方法通过一种收缩迭代求解器(如优化预处理共轭梯度法或几何多网格)来引导自适应网格细化以及线性系统的近似求解。我们以最优代数率证明了所提出的自适应算法的线性收敛性。与之前的工作不同,我们不仅考虑了与自由度数量相关的速率,甚至证明了最佳复杂性,即与总计算成本相关的最佳收敛速率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Goal-oriented adaptive finite element methods with optimal computational complexity.

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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