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Monotone and consistent discretization of the Monge-Ampère operator


Authors: Jean-David Benamou, Francis Collino and Jean-Marie Mirebeau
Journal: Math. Comp. 85 (2016), 2743-2775
MSC (2010): Primary 35J96, 65N06
DOI: https://doi.org/10.1090/mcom/3080
Published electronically: March 22, 2016
MathSciNet review: 3522969
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Abstract: We introduce a novel discretization of the Monge-Ampère operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.


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Additional Information

Jean-David Benamou
Affiliation: Mokaplan, INRIA, Domaine de Voluceau, BP 105 78153, Le Chesnay Cedex, France
Email: jean-david.benamou@inria.fr

Francis Collino
Affiliation: Mokaplan, INRIA, Domaine de Voluceau BP 105 78153, Le Chesnay Cedex, France

Jean-Marie Mirebeau
Affiliation: Laboratoire de Mathématiques d’Orsay, University Paris-Sud, CNRS, University Paris-Saclay, 91405 Orsay, France
Email: jean-marie.mirebeau@math.u-psud.fr

DOI: https://doi.org/10.1090/mcom/3080
Keywords: Monge-Amp\`ere PDE, monotone finite differences scheme, lattice basis reduction, Stern-Brocot tree
Received by editor(s): September 23, 2014
Received by editor(s) in revised form: May 11, 2015
Published electronically: March 22, 2016
Additional Notes: This work was partially supported by the ANR grant NS-LBR ANR-13-JS01-0003-01
Article copyright: © Copyright 2016 American Mathematical Society