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

Mathieu Barré; Patrick Ciarlet

doi:10.5802/crmath.590

Research article - Numerical analysis, Partial differential equations

The

T

-coercivity approach for mixed problems

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.

Abstract
Résumé

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.

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.

Received: 2022-10-19
Revised: 2023-10-20
Accepted: 2023-11-16
Published online: 2024-11-05

DOI: 10.5802/crmath.590

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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G10_1051_0,
     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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