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A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

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Abstract

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 , Berlin, Germany

    Carsten Carstensen & Dietmar Gallistl

  2. Department of Computational Science and Engineering, Yonsei University, 120-749 , Seoul, Korea

    Carsten Carstensen

  3. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, Peoples Republic of China

    Jun Hu

Authors
  1. Carsten Carstensen
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  2. Dietmar Gallistl
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  3. Jun Hu
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Corresponding authors

Correspondence to Carsten Carstensen or Jun Hu.

Additional information

Dedicated to Professor Z.-C. Shi in honour of his 80th birthday.

This project was supported by the Chinesisch-Deutsches Zentrum project GZ578. The first two authors were partially supported by the DFG Research Center MATHEON. The research of the third author was supported by the NSFC Project 10971005, partially supported by the NSFC Project 11271035 and by the NSFC Key Project 11031006.

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Carstensen, C., Gallistl, D. & Hu, J. A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles. Numer. Math. 124, 309–335 (2013). https://doi.org/10.1007/s00211-012-0513-5

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  • DOI: https://doi.org/10.1007/s00211-012-0513-5

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