A finite element method for a two-dimensional Pucci equation

Susanne C. Brenner; Li-yeng Sung; Zhiyu Tan

doi:10.5802/crmeca.224

A finite element method for a two-dimensional Pucci equation
[Méthode des éléments finis pour une équation de Pucci à deux dimensions]

Susanne C. Brenner ¹ ; Li-yeng Sung ¹ ; Zhiyu Tan ²

¹ Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA
² Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA

Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 261-276.

Résumé
Abstract

Cet article étudie une méthode d’éléments finis non linéaire des moindres carrés pour les solutions fortes du problème de la valeur limite de Dirichlet d’une équation de Pucci bidimensionnelle sur des domaines polygonaux convexes. Nous obtenons des estimations d’erreur a priori et a posteriori et présentons des résultats numériques corroborants, où les problèmes d’optimisation discrets non lisses et non linéaires sont résolus par une méthode d’ensemble active et une méthode de direction alternée avec multiplicateurs.

A nonlinear least-squares finite element method for strong solutions of the Dirichlet boundary value problem of a two-dimensional Pucci equation on convex polygonal domains is investigated in this paper. We obtain a priori and a posteriori error estimates and present corroborating numerical results, where the discrete nonsmooth and nonlinear optimization problems are solved by an active set method and an alternating direction method with multipliers.

Reçu le : 2023-02-17
Accepté le : 2023-09-06
Première publication : 2023-11-21
Publié le : 2024-04-26

DOI : 10.5802/crmeca.224

Keywords: Pucci’s equation, finite element method, strong solution, a priori and a posteriori error estimates, nonlinear least-squares, active set method, ADMM
Mot clés : équation de Pucci, méthode des éléments finis, solution forte, estimations d’erreur a priori et a posteriori, moindres carrés non linéaires, méthode des ensembles actifs, ADMM

Affiliations des auteurs :

Susanne C. Brenner ¹ ; Li-yeng Sung ¹ ; Zhiyu Tan ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Susanne C. Brenner and Li-yeng Sung and Zhiyu Tan},
     title = {A finite element method for a two-dimensional {Pucci} equation},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {261--276},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
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     year = {2023},
     doi = {10.5802/crmeca.224},
     language = {en},
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Susanne C. Brenner; Li-yeng Sung; Zhiyu Tan. A finite element method for a two-dimensional Pucci equation. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 261-276. doi : 10.5802/crmeca.224. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.224/

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