Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms

Volker Kempf

doi:10.5802/crmath.424

Numerical analysis

Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms

Volker Kempf ¹

¹Universität der Bundeswehr München, Germany

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 437-443

Abstract (Original language)
Résumé

The Brezzi–Douglas–Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in $L^{2}$ , the other considering parallelotopes with estimates in terms of $L^{p}$ -norms. This contribution provides generalized estimates for anisotropic simplices for the $L^{p}$ case, $1 \leq p \leq \infty$ , and shows new estimates for anisotropic prisms with triangular base.

L’erreur d’interpolation de Brezzi–Douglas–Marini sur les éléments anisotropes a été analysée dans deux publications récentes, la première se concentrant sur les simplices avec des estimations dans $L^{2}$ , l’autre considérant les parallelotopes avec des estimations en termes de normes $L^{p}$ . Notre contribution fournit des estimations généralisées pour les simplexes anisotropes pour le cas $L^{p}$ , $1 \leq p \leq \infty$ , et montre de nouvelles estimations pour les prismes anisotropes à base triangulaire.

Received: 2022-05-12
Revised: 2022-09-09
Accepted: 2022-09-15
Published online: 2023-01-12

DOI: 10.5802/crmath.424

Classification: 65D05, 65N30

Author's affiliations:

Volker Kempf ¹

¹ Universität der Bundeswehr München, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Volker Kempf},
     title = {Brezzi{\textendash}Douglas{\textendash}Marini interpolation on anisotropic simplices and prisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {437--443},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     doi = {10.5802/crmath.424},
     language = {en},
}

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Volker Kempf. Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 437-443. doi: 10.5802/crmath.424

References
Cited by

[1] Gabriel Acosta; Thomas Apel; Ricardo G. Durán; Ariel L. Lombardi Error estimates for Raviart–Thomas interpolation of any order on anisotropic tetrahedra, Math. Comput., Volume 80 (2011) no. 273, pp. 141-163 | MR | Zbl | DOI

[2] Thomas Apel; Volker Kempf Brezzi–Douglas–Marini interpolation of any order on anisotropic triangles and tetrahedra, SIAM J. Numer. Anal., Volume 58 (2020) no. 3, pp. 1696-1718 | MR | Zbl | DOI

[3] Thomas Apel; Volker Kempf Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading, Calcolo, Volume 58 (2021) no. 2, 15 | MR | Zbl | DOI

[4] Daniele Boffi; Franco Brezzi; Michel Fortin Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, 44, Springer, 2013 | Zbl | DOI

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[6] Volker Kempf Pressure-robust discretizations for incompressible flow problems on anisotropic meshes, Doctoral Thesis, Universität der Bundeswehr München, Deutschland (2022)

[7] Alexander Linke On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Eng., Volume 268 (2014), pp. 782-800 | MR | Zbl | DOI

[8] Jean-Claude Nédélec A new family of mixed finite elements in $ℝ^{3}$ , Numer. Math., Volume 50 (1986) no. 1, pp. 57-81 | MR | Zbl | DOI

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