Comptes Rendus
Numerical Analysis/Partial Differential Equations
On the numerical solution of a two-dimensional Pucci's equation with Dirichlet boundary conditions: a least-squares approach
Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 375-380.

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 H2(Ω)×Q, where Q is the space of the 2×2 symmetric tensor-valued functions with components in L2(Ω). After an appropriate space discretization the resulting finite dimensional problem is solved by an iterative method operating alternatively in the spaces Vh and Qh approximating H2(Ω) and Q, respectively. The results of numerical experiments are presented; they validate the methodology discussed in this Note.

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 H2(Ω)×QQ est l'espace des fonctions à valeurs tensorielles symetriques 2×2, dont les composantes sont dans L2(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 Vh et Qh, approximations respectives de H2(Ω) et Q. Les résultats d'expériences numériques sont presentés ; ils valident la méthodologie numérique décrite dans cette Note.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.08.002
Edward J. Dean 1; Roland Glowinski 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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Edward J. Dean; Roland Glowinski. On the numerical solution of a two-dimensional Pucci's equation with Dirichlet boundary conditions: a least-squares approach. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 375-380. doi : 10.1016/j.crma.2005.08.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.08.002/

[1] R.P. Brent Algorithms for Minimization without Derivatives, Dover Publications, Mineola, NY, 2002

[2] X. Cabré Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations, Discrete and Continuous Dynamical Systems, Volume 8 (2002) no. 2, pp. 289-302

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

[4] E.J. Dean; R. Glowinski 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] E.J. Dean; R. Glowinski 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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