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.

Abstract (VO)
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.

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/

Bibliographie
Cité par

[1] F. Assous; P. Ciarlet Jr.; S. Labrunie Mathematical foundations of computational electromagnetism, Applied Mathematical Sciences, 198, Springer, 2018 | DOI | MR | Zbl

[2] I. Babuška The finite element method with Lagrangian multipliers, Numer. Math., Volume 20 (1973) no. 3, pp. 179-192 | DOI | MR | Zbl

[3] A.-S. Bonnet-Ben Dhia; L. Chesnel; X. Claeys Radiation condition for a non-smooth interface between a dielectric and a metamaterial, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 09, pp. 1629-1662 | DOI | MR | Zbl

[4] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Jr. $T$ -coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM, Math. Model. Numer. Anal., Volume 46 (2012), pp. 1363-1387 | DOI | MR | Zbl

[5] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Jr. $T$ -coercivity for the Maxwell problem with sign-changing coefficients, Commun. Partial Differ. Equations, Volume 39 (2014), pp. 1007-1031 | DOI | MR | Zbl

[6] A.-S. Bonnet-Ben Dhia; L. Chesnel; P.. Ciarlet Jr. Two-dimensional Maxwell’s equations with sign-changing coefficients, Appl. Numer. Math., Volume 79 (2014), pp. 29-41 | DOI | MR | Zbl

[7] A.-S. Bonnet-Ben Dhia; C. Carvalho; P. Ciarlet Jr. Mesh requirements for the finite element approximation of problems with sign-changing coefficients, Numer. Math., Volume 138 (2018), pp. 801–-838 | DOI | MR | Zbl

[8] A.-S. Bonnet-Ben Dhia; P. Ciarlet Jr.; C. M. Zwölf Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math., Volume 234 (2010) no. 6, pp. 1912-1919 | DOI | MR | Zbl

[9] D. Boffi; F. Brezzi; M. Fortin Mixed finite element methods and applications, Springer Series in Computational Mathematics, 44, Springer, 2013 | DOI | MR | Zbl

[10] A. Buffa; S. H. Christiansen The electric field integral equation on Lipschitz screens: definitions and numerical approximation, Numer. Math., Volume 94 (2003) no. 2, pp. 229-267 | DOI | MR | Zbl

[11] A. Buffa; S. H. Christiansen A dual finite element complex on the barycentric refinement, C. R. Math., Volume 340 (2005) no. 6, pp. 461-464 | DOI | Numdam | Zbl

[12] R. Bunoiu; L. Chesnel; K. Ramdani; M. Rihani Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients, Ann. Fac. Sci. Toulouse, Math., Volume 30 (2022) no. 5, pp. 1075-1119 | DOI | MR | Zbl

[13] A. Buffa; M. Costabel; C. Schwab Boundary element methods for Maxwell’s equations on non-smooth domains, Numer. Math., Volume 92 (2002) no. 4, pp. 679-710 | DOI | MR | Zbl

[14] M. Barré; C. Grandmont; P. Moireau Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials, Evol. Equ. Control Theory, Volume 12 (2023) no. 3, pp. 846-906 | DOI | MR | Zbl

[15] M. Barré; C. Grandmont; P. Moireau Numerical analysis of an incompressible soft material poromechanics model using T-coercivity, C. R. Mécanique, Volume 351 (2023) no. S1, pp. 17-52 | DOI

[16] R. Bunoiu; K. Ramdani Homogenization of materials with sign changing coefficients, Commun. Math. Sci., Volume 14 (2016) no. 4, pp. 1137-1154 | DOI | Zbl

[17] F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Franc. Automat. Inform. Rech. Operat., R (1974) no. 2, pp. 129-151 | Numdam | MR | Zbl

[18] R. Bunoiu; K. Ramdani; C. Timofte T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains, Electron. J. Differ. Equ., Volume 2021 (2021), 59 | MR | Zbl

[19] R. Bunoiu; K. Ramdani; C. Timofte Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump, Commun. Math. Sci., Volume 21 (2023) no. 7, pp. 2029-2049 | DOI | MR | Zbl

[20] A. Buffa Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations, SIAM J. Numer. Anal., Volume 43 (2005), pp. 1-18 | DOI | MR | Zbl

[21] C. Carvalho; L. Chesnel; P. Ciarlet Jr. Eigenvalue problems with sign-changing coefficients, C. R. Math., Volume 355 (2017) no. 6, pp. 671-675 | DOI | Zbl

[22] L. Chesnel; P. Ciarlet Jr. T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math., Volume 124 (2013) no. 1, pp. 1-29 | DOI | MR | Zbl

[23] L. Chesnel Bilaplacian problems with a sign-changing coefficient, Math. Methods Appl. Sci., Volume 39 (2016) no. 17, pp. 4964-4979 | DOI | MR | Zbl

[24] P. Ciarlet Jr. $T$ -coercivity: Application to the discretization of Helmholtz-like problems, Comput. Math. Appl., Volume 64 (2012) no. 1, pp. 22-34 | DOI | MR | Zbl

[25] P. Ciarlet Jr. Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient (2020) (Technical Report HAL, https://hal.inria.fr/hal-02567484v1)

[26] P. Ciarlet Jr. Lecture notes on Maxwell’s equations and their approximation (in French), Ph. D. Thesis, Paris-Saclay University and Institut Polytechnique de Paris, Paris, France (2021) (Master’s degree Analysis, Modelling and Simulation, https://hal.inria.fr/hal-03153780)

[27] P. Ciarlet Jr. On the approximation of electromagnetic fields by edge finite elements – Part 4: analysis of the model with one sign-changing coefficient, Numer. Math., Volume 152 (2022), pp. 223-257 | DOI | MR | Zbl

[28] P. Ciarlet Jr.; E. Jamelot; F. D. Kpadonou Domain Decomposition Methods for the diffusion equation with low-regularity solution, Comput. Math. Appl., Volume 74 (2017), pp. 2369-2384 | DOI | MR | Zbl

[29] G. Duvaut; J.-L. Lions Les inéquations en mécanique et en physique, Travaux et recherches mathematiques, 21, Dunod, 1972 | MR | Zbl

[30] A. Ern; J.-L. Guermond Finite Elements II: Galerkin approximation, elliptic and mixed PDEs, Texts in Applied Mathematics, 73, Springer, 2021 | DOI | MR | Zbl

[31] A. Ern; J.-L. Guermond Finite Elements III. First-order and time-dependent PDEs, Texts in Applied Mathematics, 74, Springer, 2021 | DOI | MR | Zbl

[32] M. Fortin An analysis of the convergence of mixed finite element methods, RAIRO, Anal. Numér., Volume 11 (1977) no. 4, pp. 341-354 | DOI | Numdam | Zbl

[33] L. Giret Numerical analysis of a non-conforming Domain Decomposition for the multigroup SPN equations, Ph. D. Thesis, Paris-Saclay University, Paris, France (2018) (https://pastel.archives-ouvertes.fr/tel-01936967)

[34] V. Girault; P.-A. Raviart Finite element methods for Navier–Stokes equations: theory and algorithms, Springer Series in Computational Mathematics, 5, Springer, 1986 | DOI | MR | Zbl

[35] M. Halla Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility, Numer. Math., Volume 148 (2021) no. 2, pp. 387-407 | DOI | MR | Zbl

[36] M. Halla On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions, IMA J. Numer. Anal., Volume 43 (2023) no. 1, pp. 535-559 | DOI | MR | Zbl

[37] R. Hiptmair Finite elements in computational electromagnetics, Acta Numer. (2002), pp. 237-339 | DOI | MR | Zbl

[38] Q. Hong; J. Kraus; M. Lymbery; F. Philo A new practical framework for the stability analysis of perturbed saddle-point problems and applications, Math. Comput., Volume 92 (2023), pp. 607-634 | DOI | MR | Zbl

[39] T. Hohage; L. Nannen Convergence of infinite element methods for scalar waveguide problems, BIT, Volume 55 (2015), pp. 215-254 | DOI | MR | Zbl

[40] E. Jamelot Improved stability estimates for solving Stokes problem with Fortin-Soulie finite elements (2023) (Technical Report HAL,no. cea-03833616, https://hal-cea.archives-ouvertes.fr/cea-03833616v2)

[41] E. Jamelot; P. Ciarlet Jr. Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation, J. Comput. Phys., Volume 241 (2013), pp. 445-463 | DOI | MR | Zbl

[42] O. A. Ladyzhenskaya The mathematical theory of viscous incompressible flow, 2, Gordon and Breach New York, 1969 | MR | Zbl

[43] David P. Levadoux Analyse numérique de la formulation intégrodifférentielle d’un problème de Maxwell harmonique impliquant un diélectrique traversé de surfaces exfoliées métalliques et impédantes (2022) (Technical Report HAL, https://hal.archives-ouvertes.fr/hal-03644547)

[44] S. Nicaise; J. Venel A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients, J. Comput. Appl. Math., Volume 235 (2011), pp. 4272-4282 | DOI | Zbl

[45] F.-J. Sayas; T. S. Brown; M. E. Hassell Variational techniques for elliptic partial differential equations, CRC Press, 2019 | DOI | MR | Zbl

[46] G. Unger Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems, SN Partial Differ. Equ. Appl., Volume 2 (2021), 39 | DOI | MR | Zbl

[47] C. Weber A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci., Volume 2 (1980) no. 1, pp. 12-25 | DOI | Zbl

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