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Comptes Rendus Mathématique
Volume 341, n° 6
pages 375-380 (septembre 2005)
Doi : 10.1016/j.crma.2005.08.002
Received : 29 July 2005 ;  accepted : 8 August 2005
On the numerical solution of a two-dimensional Pucciʼs equation with Dirichlet boundary conditions: a least-squares approach
Sur la solution numérique de lʼéquation bi-dimensionelle de Pucci avec conditions limites de Dirichlet : une formulation par moindres carrés
 

Edward J. Dean a , Roland Glowinski b, a
a Department of Mathematics, University of Houston, Houston, TX 77024-3008, USA 
b Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France 

Abstract

In this Note we discuss the numerical solution of a two-dimensional, fully nonlinear elliptic equation of the Pucciʼs type, completed by Dirichlet boundary conditions. The solution method relies on a least-squares formulation taking place in a subset of  , where Q is the space of the   symmetric tensor-valued functions with components in  . After an appropriate space discretization the resulting finite dimensional problem is solved by an iterative method operating alternatively in the spaces   and   approximating   and Q, respectively. The results of numerical experiments are presented; they validate the methodology discussed in this Note. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Dans cette Note, on étudie la résolution numérique dʼune 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   où Q est lʼespace des fonctions à valeurs tensorielles symetriques  , dont les composantes sont dans  . 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   et  , approximations respectives de   et Q. Les résultats dʼexpériences numériques sont presentés ; ils valident la méthodologie numérique décrite dans cette Note. Pour citer cet article : E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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© 2005  Published by Elsevier Masson SAS de la part de Académie des sciences.
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