[Un complexe div-div discret avec sérendipité sur maillages polygonaux]
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.
Accepté le :
Première publication :
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
CC-BY 4.0
@article{CRMECA_2023__351_S1_A8_0,
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},
publisher = {Acad\'emie des sciences, Paris},
year = {2023},
doi = {10.5802/crmeca.150},
language = {en},
note = {Online first},
}
TY - JOUR AU - Michele Botti AU - Daniele A. Di Pietro AU - Marwa Salah TI - A serendipity fully discrete div-div complex on polygonal meshes JO - Comptes Rendus. Mécanique PY - 2023 PB - Académie des sciences, Paris N1 - Online first DO - 10.5802/crmeca.150 LA - en ID - CRMECA_2023__351_S1_A8_0 ER -
Michele Botti; Daniele A. Di Pietro; Marwa Salah. A serendipity fully discrete div-div complex on polygonal meshes. Comptes Rendus. Mécanique, Online first (2023), pp. 1-31. doi : 10.5802/crmeca.150.
[1] Mixed finite elements for elasticity, Numer. Math., Volume 92 (2002) no. 3, pp. 401-419 | DOI | MR | Zbl
[2] Rectangular mixed finite elements for elasticity, Math. Models Methods Appl. Sci., Volume 15 (2005) no. 9, pp. 1417-1429 | DOI | MR | Zbl
[3] Differential complexes and stability of finite element methods. II. The elasticity complex, Compatible spatial discretizations (The IMA Volumes in Mathematics and its Applications), Volume 142, Springer, 2006, pp. 47-67 | DOI | MR | Zbl
[4] Finite elements for symmetric tensors in three dimensions, Math. Comput., Volume 77 (2008) no. 263, pp. 1229-1251 | DOI | MR | Zbl
[5] Finite elements for divdiv-conforming symmetric tensors in three dimensions, Math. Comput. (2022) (Online first) | DOI | Zbl
[6] On , and finite elements for matrix-valued Curl problems (2022) (https://arxiv.org/abs/2202.08740) | DOI
[7] Finite Element Systems for Vector Bundles: Elasticity and Curvature, Found. Comput. Math. (2022) (Online first) | DOI | Zbl
[8] Constructions of some minimal finite element systems, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 833-850 | DOI | Numdam | MR | Zbl
[9] Multigrid methods for Hellan–Herrmann–Johnson mixed method of Kirchhoff plate bending problems, J. Sci. Comput., Volume 76 (2018) no. 2, pp. 673-696 | DOI | MR | Zbl
[10] A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method (I. Galligani; E. Magenes, eds.) (Lecture Notes in Mathematics), Volume 606, Springer, 1977 | DOI | MR | Zbl
[11] A fully discrete plates complex on polygonal meshes with application to the Kirchhoff–Love problem, Math. Comput., Volume 92 (2023) no. 339, pp. 51-77 | DOI | MR | Zbl
[12] Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, Math. Models Methods Appl. Sci., Volume 30 (2020) no. 9, pp. 1809-1855 | DOI | MR | Zbl
[13] An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency, Found. Comput. Math., Volume 23 (2023), pp. 85-164 | DOI | Zbl
[14] A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Eng., Volume 111 (1994) no. 3-4, pp. 283-303 | DOI | MR | Zbl
[15] Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan J. Ind. Appl. Math., Volume 12 (1995) no. 3, pp. 487-514 | DOI | MR | Zbl
[16] Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method, ESAIM, Math. Model. Numer. Anal., Volume 57 (2023) no. 1, pp. 191-225 | DOI | MR
[17] Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes, Comput. Methods Appl. Mech. Eng., Volume 397 (2022), 115061 | DOI | MR | Zbl
[18] Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., Volume 26 (2016) no. 4, pp. 729-750 | DOI | MR | Zbl
[19] Serendipity face and edge VEM spaces, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 28 (2017) no. 1, pp. 143-180 | DOI | MR | Zbl
[20] Serendipity virtual elements for general elliptic equations in three dimensions, Chin. Ann. Math., Ser. B, Volume 39 (2018) no. 2, pp. 315-334 | DOI | MR | Zbl
[21] The Hybrid High-Order method for polytopal meshes. Design, analysis, and applications, Modeling, Simulation and Applications, 19, Springer, 2020 | DOI
[22] The Hellan–Herrmann–Johnson method: some new error estimates and postprocessing, Math. Comput., Volume 52 (1989) no. 185, pp. 17-29 | DOI | MR | Zbl
[23] Finite elements for divdiv-conforming symmetric tensors (2021) (https://arxiv.org/abs/2005.01271v3)
[24] The divDiv-complex and applications to biharmonic equations, Appl. Anal., Volume 99 (2020) no. 9, pp. 1579-1630 | DOI | MR | Zbl
[25] A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comput., Volume 86 (2017) no. 307, pp. 2159-2191 | DOI | MR | Zbl
[26] Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 150 | DOI | MR | Zbl
[27] Poincaré meets Korn via Maxwell: Extending Korn’s first inequality to incompatible tensor fields, J. Differ. Equations, Volume 258 (2015) no. 4, pp. 1267-1302 | DOI | Zbl
Cité par Sources :
Commentaires - Politique
Novel transport phenomena in graphene induced by strong spin-orbit interaction
Taro Wakamura; Sophie Guéron; Hélène Bouchiat
C. R. Phys (2021)