Comptes Rendus
Numerical Analysis
Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach
Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 779-784.

The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa–Douglas–Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the H 2 (Ω)-regularity, but not the C 2 (Ω ¯) one, does not prevent optimal order of convergence. Finally, the third test problem having no smooth solution is more costly to solve and leads to discrete solutions showing negative curvature near the corners.

L'objet essentiel de cette Note est l'étude d'une méthode pour la résolution numérique du problème de Dirichlet pour l'équation de Monge–Ampère elliptique en dimension deux (le problème E-MAD). Cette méthode repose sur une reformulation de E-MAD comme un problème de Calcul des Variations impliquant l'opérateur bi-harmonique (ou des opérateurs voisins), puis sur une formulation de type point-selle pour un Lagrangien augmenté bien choisi, ce qui conduit naturellement à des algorithmes du type Uzawa–Douglas–Rachford. La méthodologie ci-dessus s'applique à des problèmes autres que E-MAD (l'équation de Pucci, par exemple). Les résultats d'essais numériques sont egalement presentés. Ils concernent la résolution du problème E-MAD sur le carré unité (0,1)×(0,1). Le premier problème test a une solution régulière (analytique, en fait) connue exactement ; on la retrouve facilement, avec une erreur d'approximation d'ordre optimal. La solution du second probleme test est aussi connue exactement ; le fait qu'elle soit dans H 2 (Ω) sans être dans C 2 (Ω ¯) n'empêche pas d'obtenir une erreur d'approximation d'ordre optimal. Finalement, le troisième problème test n'ayant pas de solution régulière est plus difficile à résoudre ; les solutions approchées obtenues montrent que la courbure devient negative au voisinage des coins.

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DOI: 10.1016/S1631-073X(03)00149-3
Edward J. Dean 1; Roland Glowinski 1

1 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
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Edward J. Dean; Roland Glowinski. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 779-784. doi : 10.1016/S1631-073X(03)00149-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00149-3/

[1] T. Aubin Nonlinear Analysis on Manifolds, Springer-Verlag, Berlin, 1982

[2] L.A. Caffarelli, The Monge–Ampère equation and optimal transportation: an elementary review, Lecture at ICM 2002, Beijing, August 20–28, 2002

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

[4] M. Fortin; R. Glowinski Augmented Lagrangians, North-Holland, Amsterdam, 1983

[5] R. Glowinski; P. Le Tallec Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989

[6] R. Glowinski; J.-L. Lions; R. Tremolieres Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981

[7] V.I. Oliker; L.D. Prussner On the numerical solution of the equation zxxzyyzxy2=f and its discretization, I, Numer. Math., Volume 54 (1988), pp. 271-293

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