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Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

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Abstract

This paper discusses convergence and complexity of arbitrary, but fixed, order adaptive mixed element methods for the Poisson equation in two and three dimensions. The two main ingredients in the analysis, namely the quasi-orthogonality and the discrete reliability, are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition. The adaptive algorithms are shown to be contractive for the sum of the error of flux in L 2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions. The methods do not require a separate treatment for the data oscillation.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    JianGuo Huang

  2. Division of Computational Science, E-Institute of Shanghai Universities, Shanghai, 200234, China

    JianGuo Huang

  3. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    YiFeng Xu

  4. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, China

    YiFeng Xu

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  1. JianGuo Huang
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  2. YiFeng Xu
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Correspondence to YiFeng Xu.

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Huang, J., Xu, Y. Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation. Sci. China Math. 55, 1083–1098 (2012). https://doi.org/10.1007/s11425-012-4384-0

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