The Dual Characteristic-Galerkin Method

Frédéric Hecht; Olivier Pironneau

doi:10.5802/crmath.598

Article de recherche - Algorithmes et outils informatiques

The Dual Characteristic-Galerkin Method
[La méthode des caractéristiques-Galerkin duale]

Frédéric Hecht ¹ ; Olivier Pironneau ¹

¹ LJLL, Boite 187, Sorbonne Université, Place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1109-1119.

Résumé
Abstract

La méthode Dual Characteristic-Galerkin (DCGM) est conservative, précise et expérimentalement positive. Nous prouvons la convergence et la stabilité $L^{2}$ . Dans le cadre numérique des méthodes d’éléments finis (FEM) en 2D, la méthode est comparée à la méthode Primal Characteristic-Galerkin (PCGM), au Streamline upwinding (SUPG), à la méthode Dual Discontinuous Galerkin (DDG) et à une discretisation FEM sans décentrage. La méthode DCGM est difficile à mettre en œuvre numériquement, mais elle est de loin supérieure à toutes les autres dans le cadre étudié dans cette note.

The Dual Characteristic-Galerkin method (DCGM) is conservative, precise and experimentally positive. We present the method and prove convergence and $L^{2}$ -stability in the case of Neumann boundary conditions. In a 2D numerical finite element setting (FEM), the method is compared to Primal Characteristic-Galerkin (PCGM), Streamline upwinding (SUPG), the Dual Discontinuous Galerkin method (DDG) and centered FEM without upwinding. DCGM is difficult to implement numerically but, in the numerical context of this note, it is far superior to all others.

Reçu le : 2023-10-23
Accepté le : 2023-12-06
Accepté après révision le : 2024-01-19
Publié le : 2024-11-05

DOI : 10.5802/crmath.598

Classification : 35Q35, 65M06, 65M15, 65M25, 65M60
Keywords: Partial differential equations, convection-diffusion, numerical method, finite element method
Mot clés : Équations aux dérivées partielles, convection-diffusion, schémas numériques, éléments finis

Affiliations des auteurs :

Frédéric Hecht ¹ ; Olivier Pironneau ¹

¹ LJLL, Boite 187, Sorbonne Université, Place Jussieu, 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fr\'ed\'eric Hecht and Olivier Pironneau},
     title = {The {Dual} {Characteristic-Galerkin} {Method}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1109--1119},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.598},
     language = {en},
}

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Frédéric Hecht; Olivier Pironneau. The Dual Characteristic-Galerkin Method. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1109-1119. doi : 10.5802/crmath.598. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.598/

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