The $\mathtt{T}$-coercivity approach for mixed problems

Mathieu Barré; Patrick Ciarlet

doi:10.5802/crmath.590

Article de recherche - Analyse numérique, Équations aux dérivées partielles

The

T

-coercivity approach for mixed problems
[

T

-coercivité et problèmes mixtes]

Mathieu Barré ^1,² ; Patrick Ciarlet ³

¹ Inria, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
² LMS, École Polytechnique, CNRS, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau, France
³ POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1051-1088.

Résumé
Abstract

Classiquement, le caractère bien posé des formulations variationnelles de problèmes linéaires mixtes est obtenu à l’aide de la condition inf-sup sur la contrainte. Dans cette note, nous proposons un cadre alternatif pour étudier de tels problèmes en utilisant la notion de $T$ -coercivité pour obtenir une condition inf-sup globale. Il s’agit d’une approche constructive qui permet en outre de concevoir simplement des approximations numériques adaptées car la dérivation de la condition inf-sup discrète uniforme découle en général directement de l’étude du problème continu. Pour appuyer notre propos, nous résolvons une série de problèmes mixtes classiques grâce à la notion de $T$ -coercivité. Entre autres, le lemme de Fortin apparaît naturellement dans l’analyse numérique des problèmes discrets.

Classically, the well-posedness of variational formulations of mixed linear problems is achieved through the inf-sup condition on the constraint. In this note, we propose an alternative framework to study such problems by using the $T$ -coercivity approach to derive a global inf-sup condition. Generally speaking, this is a constructive approach that, in addition, drives the design of suitable approximations. As a matter of fact, the derivation of the uniform discrete inf-sup condition for the approximate problems follows easily from the study of the original problem. To support our view, we solve a series of classical mixed problems with the $T$ -coercivity approach. Among others, the celebrated Fortin Lemma appears naturally in the numerical analysis of the approximate problems.

Reçu le : 2022-10-19
Révisé le : 2023-10-20
Accepté le : 2023-11-16
Publié le : 2024-11-05

DOI : 10.5802/crmath.590

Classification : 65N30, 35J57, 76D07, 78M10

Affiliations des auteurs :

Mathieu Barré ^{1, 2} ; Patrick Ciarlet ³

¹ Inria, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
² LMS, École Polytechnique, CNRS, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau, France
³ POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mathieu Barr\'e and Patrick Ciarlet},
     title = {The $\mathtt{T}$-coercivity approach for mixed problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1051--1088},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.590},
     language = {en},
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Mathieu Barré; Patrick Ciarlet. The $\mathtt{T}$-coercivity approach for mixed problems. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1051-1088. doi : 10.5802/crmath.590. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.590/

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