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Convergence analysis of the adaptive finite element method with the red-green refinement

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Abstract

In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasiorthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).

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Authors and Affiliations

  1. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    XuYing Zhao & ZhongCi Shi

  2. Key Laboratory of Mathematics and Applied Mathematics (Peking University), Ministry of Education and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jun Hu

  3. Graduate University of Chinese Academy of Sciences, Beijing, 100190, China

    XuYing Zhao

Authors
  1. XuYing Zhao
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  2. Jun Hu
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  3. ZhongCi Shi
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Correspondence to Jun Hu.

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Zhao, X., Hu, J. & Shi, Z. Convergence analysis of the adaptive finite element method with the red-green refinement. Sci. China Math. 53, 499–512 (2010). https://doi.org/10.1007/s11425-009-0200-x

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  • DOI: https://doi.org/10.1007/s11425-009-0200-x

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