On the numerical solution of a two-dimensional Pucci's equation with Dirichlet boundary conditions: a least-squares approach

Dean, Edward J.; Glowinski, Roland

doi:10.1016/j.crma.2005.08.002

Numerical Analysis/Partial Differential Equations

On the numerical solution of a two-dimensional Pucci's equation with Dirichlet boundary conditions: a least-squares approach
[Sur la solution numérique de l'équation bi-dimensionelle de Pucci avec conditions limites de Dirichlet : une formulation par moindres carrés]

Présenté par : Philippe G. Ciarlet

Dean, Edward J.¹ ; Glowinski, Roland ^1,²

¹ Department of Mathematics, University of Houston, Houston, TX 77024-3008, USA
² Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Tome 341 (2005) no. 6, pp. 375-380.

Résumé
Abstract

Dans cette Note, on étudie la résolution numérique d'une équation elliptique bi-dimensionelle, pleinement non linéaire et de type Pucci. La méthode de résolution repose sur une formulation par moindres carrés dans un sous-ensemble de $H^{2} (Ω) \times Q$ où Q est l'espace des fonctions à valeurs tensorielles symetriques $2 \times 2$ , dont les composantes sont dans $L^{2} (Ω)$ . Après approximation par éléments finis, on résoud le problème en dimension finie qui en résulte par une méthode itérative qui opère alternativement dans les espaces $V_{h}$ et $Q_{h}$ , approximations respectives de $H^{2} (Ω)$ et Q. Les résultats d'expériences numériques sont presentés ; ils valident la méthodologie numérique décrite dans cette Note.

In this Note we discuss the numerical solution of a two-dimensional, fully nonlinear elliptic equation of the Pucci's type, completed by Dirichlet boundary conditions. The solution method relies on a least-squares formulation taking place in a subset of $H^{2} (Ω) \times Q$ , where Q is the space of the $2 \times 2$ symmetric tensor-valued functions with components in $L^{2} (Ω)$ . After an appropriate space discretization the resulting finite dimensional problem is solved by an iterative method operating alternatively in the spaces $V_{h}$ and $Q_{h}$ approximating $H^{2} (Ω)$ and Q, respectively. The results of numerical experiments are presented; they validate the methodology discussed in this Note.

Reçu le : 2005-07-29
Accepté le : 2005-08-08
Publié le : 2005-09-09

DOI : 10.1016/j.crma.2005.08.002

Affiliations des auteurs :

Dean, Edward J. ¹ ; Glowinski, Roland ^{1, 2}

¹ Department of Mathematics, University of Houston, Houston, TX 77024-3008, USA
² Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

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     author = {Dean, Edward J. and Glowinski, Roland},
     title = {On the numerical solution of a two-dimensional {Pucci's} equation with {Dirichlet} boundary conditions: a least-squares approach},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--380},
     publisher = {Elsevier},
     volume = {341},
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     doi = {10.1016/j.crma.2005.08.002},
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Dean, Edward J.; Glowinski, Roland. On the numerical solution of a two-dimensional Pucci's equation with Dirichlet boundary conditions: a least-squares approach. Comptes Rendus. Mathématique, Tome 341 (2005) no. 6, pp. 375-380. doi : 10.1016/j.crma.2005.08.002. http://www.numdam.org/articles/10.1016/j.crma.2005.08.002/

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Cité par

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[3] Caffarelli, L.A.; Cabré, X. Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995

[4] Dean, E.J.; Glowinski, R. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least squares approach, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004) no. 12, pp. 887-892

[5] Dean, E.J.; Glowinski, R. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 779-784

[6] E.J. Dean, R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type, Comput. Methods Appl. Mech. Engrg., in press

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