A serendipity fully discrete div-div complex on polygonal meshes

Michele Botti; Daniele A. Di Pietro; Marwa Salah

doi:10.5802/crmeca.150

A serendipity fully discrete div-div complex on polygonal meshes
[Un complexe div-div discret avec sérendipité sur maillages polygonaux]

Michele Botti ¹ ; Daniele A. Di Pietro ² ; Marwa Salah ²

¹ MOX, Politecnico di Milano, Italy
² IMAG, Univ Montpellier, CNRS, Montpellier, France

Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 219-249.

Résumé
Abstract

Dans cet article, nous abordons la réduction des degrés de liberté de face pour le complexe de l’élasticité discrète. Plus précisément, en utilisant des techniques de sérendipité, nous développons une version réduite d’un complexe bidimensionnel qui apparaît dans la discretisation des traces du complexe de l’élasticité tridimensionnel. La clé de voûte de la construction est une nouvelle estimation des fonctions polynomiales à valeurs tensorielles symétriques en termes de leur valeur au bord. Nous prouvons de nouveaux résultats pour le complexe original et montrons que le complexe réduit a les mêmes propriétés homologiques et analytiques que celui-ci. Cet article contient également une annexe avec des preuves d’inégalités de type Poincaré–Korn pour les champs hybrides.

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré–Korn-type inequalities for hybrid fields.

Reçu le : 2022-11-08
Accepté le : 2022-11-15
Première publication : 2023-03-07
Publié le : 2024-04-26

DOI : 10.5802/crmeca.150

Classification : 74K20, 74S05, 65N30
Keywords: Discrete de Rham method, serendipity, compatible discretisations, mixed formulation, div-div complex, biharmonic equation, Kirchhoff–Love plates
Mot clés : Méthode de de Rham discrète, sérendipité, discrétisations compatibles, formulation mixte, complexe div-div, équation biharmonique, plaques de Kirchhoff–Love

Affiliations des auteurs :

Michele Botti ¹ ; Daniele A. Di Pietro ² ; Marwa Salah ²

¹ MOX, Politecnico di Milano, Italy
² IMAG, Univ Montpellier, CNRS, Montpellier, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Michele Botti and Daniele A. Di Pietro and Marwa Salah},
     title = {A serendipity fully discrete div-div complex on polygonal meshes},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {219--249},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S1},
     year = {2023},
     doi = {10.5802/crmeca.150},
     language = {en},
}

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Michele Botti; Daniele A. Di Pietro; Marwa Salah. A serendipity fully discrete div-div complex on polygonal meshes. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 219-249. doi : 10.5802/crmeca.150. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.150/

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