A serendipity fully discrete div-div complex on polygonal meshes

Abstract

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.Résumé. 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.

An exterior calculus framework for polytopal methods

“…For all ℎ ∈ ,ℎ , by the definitions (4. 19) and (4.20) of the reduction and extension, it holds ℎ ℎ ℎ − ℎ ∈ ,ℎ,♭ . The proof then continues as in point 4. of the proof of Theorem 10 (see Section 3.5) with ,ℎ,♭ replaced by ,ℎ,♭ and Lemma 19 replaced by Lemma 27.…”

“…Despite their non-conformity, polytopal technologies can be used to develop compatible frameworks. Polytopal discretisations of the de Rham complex (1.1) have been proposed, e.g., in [10,33,38], and applied to a variety of models , such as magnetostatics [8,34], the Stokes equations [11], and the Yang-Mills equations [47]; they have also inspired further developments, based on the same principles, for other complexes of interest such as variants of the de Rham complex with increased regularity [32,55], elasticity complexes [19,44], and the Stokes complex [12,14,49]. Polytopal complexes have additionally been used to construct methods that are robust with respect to the variations of physical parameters, in particular for the Stokes problem [11], for the Reissner-Mindlin equation [43], or the Brinkman model [33].…”