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Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems

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Abstract

Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe (in Numer Math 103:251–266, 2006). In this paper, we extend the result to some nonsymmetric and indefinite problems. The main tools in our analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric and indefinite bilinear form. An appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order Crouzeix–Raviart element. It is shown that the ANFEM is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops. Moreover, quasi-optimality in the sense of quasi-optimal algorithmic complexity can be shown for the ANFEM. The results of numerical experiments confirm the theoretical findings.

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Authors and Affiliations

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

    Huangxin Chen & Xuejun Xu

  2. Institut für Mathematik, Universität Augsburg, 86159, Augsburg, Germany

    Ronald H. W. Hoppe

  3. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Ronald H. W. Hoppe

Authors
  1. Huangxin Chen
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  2. Xuejun Xu
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  3. Ronald H. W. Hoppe
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Correspondence to Huangxin Chen.

Additional information

Ronald H. W. Hoppe has been supported by the NSF under grants No. DMS-0511624, DMS-0707602, DMS-0810176, DMS-0811153, DMS-0914788. Xuejun Xu has been supported by the special funds for major state basic research projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (10731060).

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Chen, H., Xu, X. & Hoppe, R.H.W. Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems. Numer. Math. 116, 383–419 (2010). https://doi.org/10.1007/s00211-010-0307-6

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  • DOI: https://doi.org/10.1007/s00211-010-0307-6

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