Comptes Rendus
Numerical Analysis
Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach
Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 887-892.

We addressed, in a previous note [C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784], the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation, namely: detD2ψ=f in Ω, ψ=g on ∂Ω (ΩR2 and f>0, here). The method discussed previously relies on an augmented Lagrangian algorithm operating in the space H2(Ω) and related functional spaces of symmetric tensor-valued functions. In the particular case where the above problem has no solution in H2(Ω), while the data f and g verify {f,g}L1(Ω)×H3/2(Ω), there is strong evidence that the augmented Lagrangian algorithm discussed in previously converges-in some sense-to a least squares solution belonging to Vg={φ|φH2(Ω),φ=g on Ω}. Our goal in this note is to discuss a least-squares based alternative solution method for the Monge–Ampère Dirichlet problem. This method relies on the minimization on the set Vg×Qf (with Qf={q|q=(qij)1i,j2,qijL2(Ω),i,j,1i,j2,q=qt,detq=f}) of a well-chosen least-squares functional. From a practical point of view we solve the above minimization problem via a relaxation type algorithm, operating alternatively in Vg and Qf and very easy to combine to the mixed finite element approximations employed in the earlier work. Numerical experiments show that the above method has good convergence properties when the Monge–Ampère Dirichlet problem has solutions in Vg; they show also that, for cases where the above problem has no solution in Vg, while neither Vg nor Qf are empty, the new method reproduces the solutions obtained via the augmented Lagrangian approach, but faster.

La résolution numérique du problème de Dirichlet pour l'équation de Monge–Ampère elliptique bi-dimensionelle, soit : detD2ψ=f in Ω, ψ=g on ∂Ω (ici, ΩR2 et f>0), a été étudiée dans une note précédente [C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784]. La méthode décrite là, repose sur un algorithme de Lagrangien augmenté opérant dans l'espace H2(Ω) et des espaces associés de fonctions à valeurs tensorielles symétriques. Dans les cas où le problème ci-dessus n'a pas de solution dans H2(Ω), alors que les données f and g verifient {f,g}L1(Ω)×H3/2(Ω), diverses observations et analogies suggèrent fortement que l'algorithme de Lagrangien augmenté décrit dans notre note précédente converge-en un certain sens-vers une solution appartenant à Vg={φ|φH2(Ω),φ=g on Ω} et du type moindres carrés. L'objet de cette note est la résolution du problème de Monge–Ampère Dirichlet, directement par une méthode de moindres carrés. Cette méthode repose sur la minimisation sur l'ensemble Vg×Qf (avec Qf={q|q=(qij)1i,j2,qijL2(Ω),i,j,1i,j2,q=qt,detq=f}), d'une fonction coût bien choisie, de type moindres carrés. D'un point de vue pratique, on résout le problème de minimisation ci-dessus par un algorithme de type relaxation qui opère alternativement dans Vg et Qf ; cet algorithme est facile à combiner aux approximations par élements finis mixtes utilisées dans la note précédente. Des essais numériques montrent que la méthode de moindres carrés ci-dessus a de bonnes propriétés de convergence quand le problème de Monge–Ampère Dirichlet a des solutions dans Vg ; ces essais montrent également que lorsque problème ci-dessus n'a pas de solution dans Vg, bien que Vg et Qf soient non vides, la nouvelle méthode reproduit les solutions obtenues par Lagrangien augmenté, mais ce plus rapidement.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.018
Edward J. Dean 1; Roland Glowinski 1, 2

1 Department of Mathematics, University of Houston, Houston, Texas 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. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 887-892. doi : 10.1016/j.crma.2004.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.018/

[1] 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

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

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

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

[5] E.J. Dean, R. Glowinski, T.W. Pan, Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge–Ampère equation, in: Proceedings of the IFIP-WG7.2 Conference, Nice, 2003, Kluwer, in press

[6] R. Glowinski Finite element methods for incompressible viscous flow (P.G. Ciarlet; J.L. Lions, eds.), Handbook of Numerical Analysis, vol. IX, North-Holland, Amsterdam, 2003, pp. 3-1176

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

[8] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984

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