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Error reduction, convergence and optimality of an adaptive mixed finite element method

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Abstract

This paper proves the error reduction property (saturation property), convergence and optimality of an adaptive mixed finite element method (AMFEM) for the Poisson equation. In each step of AMFEM, the local refinement is performed basing on simple either edge-oriented residuals or edge-oriented data oscillations, depending only on the marking strategy, under some restriction of refinement. The main tools used here are the strict discrete local efficiency property given by Carstensen and Hoppe (2006) and the quasi-orthogonality estimate proved by Chen, Holst, and Xu (2009). Numerical experiments fully confirm the theoretical analysis.

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Authors and Affiliations

  1. School of Science, Chongqing Jiaotong University, Chongqing, 400074, China

    Shaohong Du

  2. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Xiaoping Xie

  3. Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, China

    Xiaoping Xie

Authors
  1. Shaohong Du
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  2. Xiaoping Xie
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Correspondence to Shaohong Du.

Additional information

This work was supported in part by the Natural Science Foundation of China under Grant No. 10771150, the National Basic Research Program of China under Grant No. 2005CB321701, and the Natural Science Foundation of Chongqing City under Grant No. CSTC, 2010BB8270.

This paper was recommended for publication by Editor Ningning YAN.

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Du, S., Xie, X. Error reduction, convergence and optimality of an adaptive mixed finite element method. J Syst Sci Complex 25, 195–208 (2012). https://doi.org/10.1007/s11424-012-9242-1

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  • DOI: https://doi.org/10.1007/s11424-012-9242-1

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