Crouzeix–Raviart and Raviart–Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

Abstract

We investigate the piecewise linear nonconforming Crouzeix–Raviart and the lowest order Raviart–Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix–Raviart and the Raviart–Thomas finite-element approximate problems. We next present the equivalence between the Raviart–Thomas finite-element method and the enriched Crouzeix–Raviart finite-element method. We emphasize that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

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Notes

  1. 1.

    Note added in proof. [16,Theorem 2] is incorrect. See the erratum in https://arxiv.org/abs/2002.09721

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP16H03950. We would like to thank the anonymous referee for the valuable comments.

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Correspondence to Hiroki Ishizaka.

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Ishizaka, H., Kobayashi, K. & Tsuchiya, T. Crouzeix–Raviart and Raviart–Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition . Japan J. Indust. Appl. Math. (2021). https://doi.org/10.1007/s13160-020-00455-7

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Keywords

  • Finite element
  • Raviart–Thomas
  • Crouzeix–Raviart
  • Anisotropic meshes

Mathematics Subject Classification

  • 65D05
  • 65N30