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

Mirebeau, Jean-Marie

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

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.

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

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