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01-11-2022

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

Author: Hiroki Ishizaka

Published in: Calcolo | Issue 4/2022

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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.

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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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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

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

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

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

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)

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

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

10.

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

11.

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

12.

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

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRef

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)

Title: Anisotropic Raviart–Thomas interpolation error estimates using a new geometric parameter
Author: Hiroki Ishizaka
Publication date: 01-11-2022
Publisher: Springer International Publishing
Published in: Calcolo / Issue 4/2022
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-022-00494-1

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