Discretization of the 3d monge−ampere operator, between wide stencils and power diagrams

Mirebeau, Jean-Marie¹ 

¹ CNRS, University Paris Dauphine, UMR 7534, Laboratory CEREMADE, Paris, France

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1511-1523.

Résumé

We introduce a monotone (degenerate elliptic) discretization of the Monge−Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded. Our approach enjoys the simplicity of the Wide Stencil method [B.D. Froese and A.M. Oberman, J. Comput. Phys. 230 (2011) 818–834.], but significantly improves its accuracy using ideas from discretizations of optimal transport based on power diagrams [F. Aurenhammer, F. Hoffmann and B. Aronov, Algorithmica (1998)]. We establish the global convergence of a damped Newton solver for the discrete system of equations. Numerical experiments, in three dimensions, illustrate the scheme efficiency.

Détail

Reçu le : 2014-11-04
DOI : 10.1051/m2an/2015016

Classification : 65N06, 65N12, 35B50, 35J60, 49L25
Keywords: Viscosity solutions, monotone numerical scheme, finite difference methods, Monge−Ampere operator

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