Skip to main content
SpringerLink
Log in

Anisotropic Raviart–Thomas interpolation error estimates using a new geometric parameter

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

We present precise Raviart–Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart–Thoma error estimates using the parameter. We also include corrections to an error in “General theory of interpolation error estimates on anisotropic meshes” (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Acosta, G., Durán, R.G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal 37, 18–36 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Acosta, G., Apel, Th., Durán, R.G., Lombardi, A.L.: Error estimates for Raviart–Thomas interpolation of any order on anisotropic tetrahedra. Math. Comput. 80(273), 141–163 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apel, Th.: Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics, Teubner, Stuttgart (1999)

    MATH  Google Scholar 

  4. Apel, Th., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47, 277–293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, New York (2013)

    Book  MATH  Google Scholar 

  7. Brandts, J., Korotov, S., Kížek, M.: On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions. Comput. Math. Appl. 55, 2227–2233 (2008)

  8. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  9. Ciarlet, P.G.: The Finite Element Method for Elliptic problems. SIAM, New York (2002)

    Book  MATH  Google Scholar 

  10. Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34, 441–463 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    Book  MATH  Google Scholar 

  12. Ern, A., Guermond, J.L.: Finite Elements I: Galerkin Approximation. Elliptic and Mixed PDEs. Springer, New York (2021)

    Book  MATH  Google Scholar 

  13. Ishizaka, H.: Anisotropic interpolation error analysis using a new geometric parameter and its applications. Ehime University, Ph. D. thesis (2022)

  14. Ishizaka, H., Kobayashi, K., Tsuchiya, T.: General theory of interpolation error estimates on anisotropic meshes. Jpn. J. Ind. Appl. Math. 38(1), 163–191 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ishizaka, H., Kobayashi, K., Tsuchiya, T.: Crouzeix–Raviart and Raviart–Thomas finite element error analysis on anisotropic meshes violating the maximum-angle condition. Jpn. J. Ind. Appl. Math. 38(2), 645–675 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ishizaka, H., Kobayashi, K., Suzuki, R., Tsuchiya, T.: A new geometric condition equivalent to the maximum angle condition for tetrahedrons. Comput Math Appl 99, 323–328 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ishizaka, H., Kobayashi, K., Tsuchiya, T.: Anisotropic interpolation error estimates using a new geometric parameter. Jpn. J. Ind. Appl. Math. 39(2) (2022)

  18. Kížek, M.: On semiregular families of triangulations and linear interpolation. Appl. Math. Praha 36, 223–232 (1991)

    Article  MathSciNet  Google Scholar 

  19. Kížek, M.: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513–520 (1992)

  20. Raviart, P. A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, I. Galligani, E. Magenes, eds., Lectures Notes in Math. 606, Springer Verlag (1977)

Download references

Acknowledgements

We would like to thank the anonymous referee for the valuable comments.

Author information

Authors and Affiliations

  1. Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan

    Hiroki Ishizaka

Authors
  1. Hiroki Ishizaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroki Ishizaka.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ishizaka, H. Anisotropic Raviart–Thomas interpolation error estimates using a new geometric parameter. Calcolo 59, 50 (2022). https://doi.org/10.1007/s10092-022-00494-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-022-00494-1

Keywords

Mathematics Subject Classification

Search

Navigation