Convergence rate estimates for Aleksandrov's solution to the MongeAmpere equation ^ast

In this paper, we establish convergence rate estimates for convex solutions to the Dirichlet problem of the Monge-Ampere equation det D ² u = f in Omega, where f is a positive and continuous function and Omega is a bounded convex domain in the Euclidean space BbbR ⁿ . We approximate the solution u by a sequence of convex polyhedra vh, which are generalized solutions to the Monge-Ampere equation in the sense of Aleksandrov, and the associated Monge-Ampere measures nu h are supported on a properly chosen grid in Omega . We will derive the convergence rate estimates for the cases when f is smooth, H"older continuous, and merely continuous.