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Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements

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Abstract

In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first-order mixed Raviart–Thomas element and the mixed Hellan–Herrmann–Johnson element, respectively. This in particular allows for proving a full one-order superconvergence result for these two mixed finite elements. Finally, a full one-order superconvergence result of both the Crouzeix-Raviart element and the Morley element follows from their special relations with the first-order mixed Raviart–Thomas element and the mixed Hellan–Herrmann–Johnson element respectively. Those superconvergence results are also extended to mildly structured meshes.

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Funding

The authors were supported by NSFC projects 11625101.

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Jun Hu

  2. Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania, PA, 16802, USA

    Limin Ma

  3. Institut für Mathematik, Humboldt-Universität zu Berlin, 10099, Berlin, Germany

    Rui Ma

Authors
  1. Jun Hu
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  2. Limin Ma
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  3. Rui Ma
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Correspondence to Limin Ma.

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Communicated by: Paul Houston

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Hu, J., Ma, L. & Ma, R. Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements. Adv Comput Math 47, 52 (2021). https://doi.org/10.1007/s10444-021-09874-7

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  • DOI: https://doi.org/10.1007/s10444-021-09874-7

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